Library 


THEORY  AND  CALCULATIONS 

OF 

ELECTRICAL  APPARATUS 


"Ms  Qraw-Ml  Book  &  Ine. 

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THEORY  AND  CALCULATIONS 

OF 

ELECTRICAL  APPARATUS 


BY 

CHARLES  PROTEUS  STEINMETZ,  A.  M.,  PH.  D. 

H 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC, 
239  WEST  39TH  STREET.     NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

6  &  8  BOUVERIE  ST.,  E.  C. 
1917 


7 


Library 


COPYRIGHT,  1917,  BY  THE 
McGRAw-HiLL  BOOK  COMPANY,  INC. 


THE  MAPLE  PRESS  YORK  PA 


PREFACE 

In  the  twenty  years  since  the  first  edition  of  "  Theory  and  Cal- 
culation of  Alternating  Current  Phenomena"  appeared,  elec- 
trical engineering  has  risen  from  a  small  beginning  to  the  world's 
greatest  industry;  electricity  has  found  its  field,  as  the  means  of 
universal  energy  transmission,  distribution  and  supply,  and  our 
knowledge  of  electrophysics  and  electrical  engineering  has  in- 
creased many  fold,  so  that  subjects,  which  twenty  years  ago  could 
be  dismissed  with  a  few  pages  discussion,  now  have  expanded 
and  require  an  extensive  knowledge  by  every  electrical  engineer. 
In  the  following  volume  I  have  discussed  the  most  important 
characteristics  of  the  numerous  electrical  apparatus,  which  have 
been  devised  and  have  found  their  place  in  the  theory  of  electrical 
engineering.  While  many  of  them  have  not  yet  reached  any 
industrial  importance,  experience  has  shown,  that  not  infre- 
quently apparatus,  which  had  been  known  for  many  years  but 
had  not  found  any  extensive  practical  use,  become,  with  changes 
of  industrial  conditions,  highly  important.  It  is  therefore 
necessary  for  the  electrical  engineer  to  be  familiar,  in  a  general 
way,  with  the  characteristics  of  the  less  frequently  used  types 
of  apparatus. 

In  some  respects,  the  following  work,  and  its  companion  vol- 
ume, "  Theory  and  Calculation  of  Electric  Circuits,"  may  be 
considered  as  continuations,  or  rather  as  parts  of  Theory  and 
Calculation  of  Alternating  Current  Phenomena."  With  the  4th 
edition,  which  appeared  nine  years  ago,  "Alternating  Current 
Phenomena"  had  reached  about  the  largest  practical  bulk,  and 
when  rewriting  it  recently  for  the  5th  edition,  it  became  necessary 
to  subdivide  it  into  three  volumes,  to  include  at  least  the  most 
necessary  structural  elements  of  our  knowledge  of  electrical 
engineering.  The  subject  matter  thus  has  been  distributed  into 
three  volumes:  " Alternating  Current  Phenomena,"  " Electric 
Circuits,"  and  "  Electrical  Apparatus." 

CHAKLES  PROTEUS  STEINMETZ. 

CAMP  MOHAWK,  VIELE'S  CREEK, 
July,  1917. 


36151 


CONTENTS 

PAGE 

PREFACE. v 

CHAPTER  I. — SPEED  CONTROL  OF  INDUCTION  MOTORS. 

I.  Starting  and  Acceleration 

1.  The  problems  of  high  torque  over  wide  range  of  speed,  and  of 
constant  speed  over  wide  range  of  load — Starting  by  armature 
rheostat 1 

2.  A.  Temperature  starting  device — Temperature  rise  increasing 
secondary  resistance  ^with  increase  of  current — Calculation  of 
motor      2 

3.  Calculation  of  numerical  instance — Its  discussion — Estimation 

of  required  temperature  rise 4 

4.  B.  Hysteresis  starting  device — Admittance  of  a  closed  mag- 
netic circuit  with  negligible  eddy  current  loss — Total  secondary 
impedance  of  motor  with  hysteresis  starting  device 5 

5.  Calculation  of  numerical  instance — Discussion — Similarity  of 
torque  curve  with  that  of  temperature  startin  gdevice — Close 

speed  regulation — Disadvantage  of  impairment  of  power  factor 
and  apparent  efficiency,  due  to  introduction  of  reactance — Re- 
quired increase  of  magnetic  density 6 

6.  C.  Eddy  current  starting  device — Admittance  of  magnetic  cir- 
cuit with  high  eddy  current  losses  and  negligible  hysteresis — 
Total  secondary  impedance  of  motor  with  eddy  current  starting 
device — Numerical  instance 8 

7.  Double  maximum  of  torque  curve — Close  speed  regulation — 
High  torque  efficiency — Poor  power  factor,  requiring  increase 
of  magnetic  density  to  get  output — Relation  to  double  squirrel 
cage  motor  and  deep  bar  motor 10 

II.  Constant  Speed  Operation 

8.  Speed  control  by  armature  resistance — Disadvantage  of  in- 
constancy of  speed  with  load — Use  of  condenser  in  armature  or 
secondary — Use  of  pyro-electric  resistance 12 

9.  Speed  control  by  variation  of  the   effective   frequency:  con- 
catenation— By  changing  the  number  of  poles:  multispeed 
motors 13 

10.  A.  Pyro-electric     speed     control — Characteristic     of     pyro- 
electric  conductor — Close  speed  regulation  of  motor — Limita- 
tion of  pyro-electric  conductors 14 

11.  B.  Condenser  speed  control — Effect  of  condenser  in  secondary, 

vii 


viii  CONTENTS 

PAGE 

giving  high  current  and  torque  at  resonance  speed — Calcula- 
tion of  motor 16 

12.  Equations   of  motor — Equation  of  torque — Speed   range  of 
maximum  torque 17 

13.  Numerical  instance — Voltampere  capacity  of  required  con- 
denser.  18 

14.  C.  Multispeed  motors — Fractional  pitch  winding,  and  switch- 
ing of  six  groups  of  coils  in  each  phase,  at  a  change  of  the  num- 
ber of  poles 20 

15.  Discussion  of  the  change  of  motor  constants  due  to  a  change  of 
the  number  of  poles,  with  series  connection  of  all  primary  turns 
— Magnetic  density  and  inferior  performance  curves  at  lower 
speeds 21 

16.  Change  of  constants  for  approximately  constant  maximum 
torque  at  all  speeds — Magnetic  density  and  change  of  coil 
connection 22 

17.  Instance  of  4  -*•  6  -f-  8  pole  motor — Numerical  calculation  and 
discussion 23 

CHAPTER  II.     MULTIPLE  SQUIRREL  CAGE  INDUCTION  MOTOR. 

18.  Superposition  of  torque  curves  of  high  resistance  low  reactance, 
and  low  resistance  high  reactance  squirrel  cage  to  a  torque 
curve  with  two  maxima,  at  high  and  at  low  speed 27 

19.  Theory  of  multiple  squirrel  cage  based  on  the  use  of  the  true 
induced  voltage,    corresponding   to  the  resultant   flux  which 
passes  beyond  the  squirrel  cage — Double  squirrel  cage  induc- 
tion motor 28 

20.  Relations  of  voltages  and  currents  in  the  double  squirrel  cage 
induction  motor 29 

21.  Equations,  and  method  of  calculation 30 

22.  Continued:  torque  and  power  equation 31 

23.  Calculation   of  numerical   instance   of   double   squirrel   cage 
motor,  speed  and  load  curves — Triple  squirrel  cage  induction 
motor 32 

24.  Equation  between  the  voltages  and  currents  in   the  triple 
squirrel  cage  induction  motor 34 

25.  Calculation  of  voltages  and  currents 35 

26."  Equation  of  torque  and  power  of  the  three  squirrel  cages,  and 

their  resultant 37 

27.  Calculation  of  numerical  instance  of  triple  squirrel  cage  induc- 
tion motor — Speed  and  load  curves 37 

CHAPTER  III.     CONCATENATION. 

Cascade  or  Tandem  Control  of  Induction  Motors 

28.  Synchronizing  of  concatenated  couple  at  half  synchronism — 
The  two  speeds  of  a  couple  of  equal  motors  and  the  three 


CONTENTS  ix 

PAGE 

speeds  of  a  couple  of  unequal  motors — Internally  concatenated 
motor 40 

29.  Generator  equation  of  concatenated  couple  above  half  syn- 
chronism— Second  range  of  motor  torque  near  full  synchron- 
ism— Generator    equation    above    full    synchronism — Ineffi- 
ciency   of   second    motor   speed    range — Its   suppression   by 
resistance  in  the  secondary  of  the  second  motor 41 

30.  General  equation  and  calculation  of  speed  and  slip  of  con- 
catenated couple 42 

31.  Calculation  of  numerical  instances 44 

32.  Calculation  of  general  concatenated  couple 45 

33.  Continued 46 

34.  Calculation  of  torque  and  power  of  the  two  motors,  and  of  the 
couple 47 

35.  Numerical  instance 48 

36.  Internally  concatenated  motor — Continuation  of  windings  into 
one  stator  and  one  rotor  winding — Fractional  pitch — No  inter- 
ference of  magnetic   flux  required — Limitation  of  available 
speed — Hunt  .motor 49 

37.  Effect  of  continuation  of  two  or  more  motors  on  the  character- 
istic constant  and  the  performance  of  the  motor 50 

CHAPTER  IV.     INDUCTION  MOTOR  WITH  SECONDARY  EXCITATION. 

38.  Large  exciting  current  and  low  power  factor  of  low  speed  in- 
duction  motors  and   motors    of    high    overload    capacity — 
Instance 52 

39.  Induction  machine  corresponding  to  synchronous  machine  ex-  • 
cited  by  armature  reaction,  induction  machine  secondary  corre- 
sponding to  synchronous  machine  field — Methods  of  secondary 
excitation :  direct  current,  commutator,  synchronous  machine, 
commutating  machine,  condenser 53 

40.  Discussion  of  the  effect  of  the  various  methods  of  secondary 
excitation  on  the  speed  characteristic  of  the  induction  motor   .     55 

Induction  Motor  Converted  to  Synchronous 

41.  Conversion  of  induction  to  synchronous  motor — Relation  of 
exciting  admittance  and  self -inductive  impedance  as  induction 
motor,  to  synchronous  impedance  and  coreloss  as  synchronous 
motor — Danielson  motor • 57 

42.  Fundamental  equation  of  synchronous  motor — Condition  of 
unity  power  factor — Condition  of  constant  field  excitation   .    .     60 

43.  Equations  of  power  input  and  output,  and  efficiency    ....     61 

44.  Numerical  instance  of  standard  induction  motor  converted  to 
synchronous — Load  curves  at  unity  power  factor  excitation  and 

at  constant  excitation 62 

45.  Numerical  instance  of  low  speed  high  excitation  induction 
motor  converted  to  synchronous  motor — Load  curves  at  unity 


CONTENTS 

PAGE 

power  factor  and  at  constant  field  excitation — Comparison 
with  induction  motor 67 

46.  Comparison  of  induction  motor  and  synchronous  motor  regard- 
ing   armature    reaction    and    synchronous    impedance — Poor 
induction  motor  makes  good,  and  good  induction  motor  makes 
poor  synchronous  motor 69 

Induction  Motor  Concatenated  with  Synchronous 

47.  Synchronous  characteristic  and  synchronizing  speed  of  con- 
catenated couple — Division  of  load  between  machines — The 
synchronous  machine  as  small  exciter 71 

48.  Equation  of  concatenated  couple  of  synchronous  and  induction 
motor — Reduction  to  standard  synchronous  motor  equation   .      72 

49.  Equation  of  power  output  and  input  of  concatenated  couple   .      74 

50.  Calculation  of  numerical  instance  of  56  polar  high  excitation 
induction  motor  concatenated  to  4  polar  synchronous   ....      75 

51.  Discussion.     High    power   factor   at    all   loads,    at    constant 
synchronous  motor  excitation 76 

Induction  Motor  Concatenated  with  Commutating  Machine 

52.  Concatenated  couple  with  commutating  machine  asynchronous 
— Series  and  shunt  excitation — Phase  relation  adjustable — 
Speed  control  and  power  factor  control — Two  independent 
variables  with  concatenated  commutating  machine,  against  one 
with  synchronous  machine — Therefore  greater  variety  of  speed 
and  load  curves 78 

53.  Representation  of  the  commutating  machine  by  an  effective 
impedance,  in  which  both  components  may  be  positive  or 
negative,  depending  on  position  of  commutator  brushes   ...      80 

54.  Calculation  of  numerical  instance,  with  commutating  machine 
series  excited  for  reactive  anti-inductive  voltage — Load  curves 
and  their  discussion 82 

Induction  Motor  with  Condenser  in  Secondary  Circuit 

55.  Shunted   capacity   neutralizing  lagging  current   of  induction 
motor — Numerical  instance — Effect  of  wave  shape  distortion — 
Condenser  in  tertiary  circuit  of  single-phase  induction  motor — 
Condensers  in  secondary  circuit — Large  amount  of  capacity 
required  by  low  frequency 84 

56.  Numerical  instance  of  low  speed  high  excitation  induction 
motor  with  capacity  in  secondary — Discussion  of  load  curves 
and  of  speed 86 

57.  Comparison  of  different  methods  of  secondary  excitation,  by 
power  factor  curves :  low  at  all  loads ;  high  at  all  loads,  low  at 
light,  high  at  heavy  loads — By  speed:  synchronous  or  constant 
speed  motors  and  asynchronous  motors  in  which  the  speed 
decreases  with  increasing  load 88 


CONTENTS  xi 

Induction  Motor  with  Commutator 

PAGE 

58.  Wave  shape  of  commutated  full  frequency  current  in  induction 
motor  secondary — Its   low   frequency   component — Full   fre- 
quency reactance  for  rotor  winding — The  two  independent 
variables :  voltage  and  phase — Speed  control  and  power  factor 
correction,  depending  on  brush  position 89 

59.  Squirrel  cage  winding  combined  with  commutated  winding — 
Heyland  motor — Available  only  for  power  factor  control — Its 
limitation 91 

CHAPTER  V.     SINGLE-PHASE  INDUCTION  MOTOR. 

60.  Quadrature  magnetic  flux  of  single-phase  induction  motor  pro- 
duced by  armature  currents — The  torque  produced  by  it — 
The  exciting  ampere-turns  and  their  change  between  synchron- 
ism and  standstill 93 

61.  Relations  between  constants  per  circuit,  and  constants  of  the 
total  polyphase  motor — Relation  thereto  of  the  constants  of 
the  motor  on  single-phase  supply — Derivation  of  the  single- 
phase  motor  constants  from  those  of  the  motor  as  three-phase  or 
quarter-phase  motor 94 

62.  Calculation  of  performance  curves  of  single-phase  induction 
motor — Torque  and,  power      96 

63.  The  different  methods  of  starting  single-phase  induction  motors 
— Phase  splitting  devices;  inductive  devices;  monocyclic  de- 
vices; phase  converter     96 

64.  Equations  of  the  starting  torque,  starting  torque  ratio,  volt- 
ampere  ratio  and  apparent  starting  torque  efficiency  of  the 
single-phase  induction  motor  starting  device 98 

65.  The  constants  of  the  single-phase  induction  motor  with  starting 
device 100 

66.  The  effective  starting  impedance  of  the  single-phase  induction 
motor — Its  approximation — Numerical  instance 101 

67.  Phase  splitting  devices — Series  impedances  with  parallel  con- 
nections of  the  two  circuits  of  a  quarter-phase  motor — Equa- 
tions     10.3 

68.  Numerical  instance  of  resistance  in  one  motor  circuit,  with 
motor  of  high  and  of  low  resistance  armature 104 

69.  Capacity  and  inductance  as  starting  device — Calculation  of 
values  to  give  true  quarter-phase  relation 106 

70.  Numerical  instance,  applied  to  motor  of  low,  and  of  high  arma- 
ture resistance 108 

71.  Series  connection  of  motor  circuits  with  shunted  impedance — 
Equations,    calculations   of   conditions   of   maximum   torque 
ratio — Numerical  instance 109 

72.  Inductive  devices — External  inductive  devices — Internal  in- 
ductive devices Ill 

73.  Shading  coil — Calculations  of  voltage  ratio  and  phase  angle   .    112 


xii  CONTENTS 

PAGE 

74.  Calculations  of  voltages,  torque,  torque  ratio  and  efficiency   .    .    114 

75.  Numerical  instance  of  shading  coil  of  low,  medium  and  high 
resistances,  with  motors  of  low,  medium  and  high  armature 
resistance 116 

76.  Monocyclic  starting  device — Applied  to  three-phase  motor — 
Equations  of  voltages,  currents,  torque,  and  torque  efficiency    .    117 

77.  Instance  of  resistance  inductance  starting  device,  of  condenser 
motor,  and  of  production  of  balanced  three-phase  triangle  by 
capacity  and  inductance 120 

78.  Numerical  instance  of  motor  with  low  resistance,  and  with 
high  resistance  armature — Discussion  of  acceleration    ....    121 

CHAPTER  VI.     INDUCTION  MOTOR  REGULATION  AND  STABILITY. 
1.    Voltage  Regulation  and  Output 

79.  Effect  of  the  voltage  drop  in  the  line  and  transformer  im- 
pedance on  the  motor — Calculation  of  motor  curves  as  affected 

by  line  impedance,  at  low,  medium  and  high  line  impedance   .    123 

80.  Load  curves  and  speed  curves — Decrease  of  maximum  torque 
and  of  power  factor  by  line  impedance — Increase  of  exciting 
current  and  decrease  of  starting  torque — Increase  of  resistance 
required  for  maximum  starting  torque 126 

2.  Frequency  Pulsation 

81.  Effect  of  frequency  pulsation — Slight  decrease  of  maximum 
torque — Great  increase  of  current  at  light  load   .......    131 

3.  Load  and  Stability 

82.  The  two  motor  speed  at  constant  torque  load — One  unstable 
and  one  stable  point — Instability  of  motor,  on  constant  torque 
load,  below  maximum  torque  point 132 

83.  Stability  at  all  speeds,  at  load  requiring  torque  proportional  to 
square    of   speed:  ship    propellor,    centrifugal    pump — Three 
speeds  at  load  requiring  torque  proportional  to  speed — Two 
stable  and  one  unstable  speed — The  two  stable  and  one  un- 
stable branch  of  the  speed  curve  on  torque  proportional  to 
speed 134 

84.  Motor  stability  function  of  the  character  of  the  load — General 
conditions  of  stability  and  instability  — Single-phase  motor   .    .    136 

4.  Generator  Regulation  and  Stability 

85.  Effect  of  the  speed  of  generator  regulation  on  maximum  output 
of  induction  motor,  at  constant  voltage — Stability  coefficient 

of  motor— Instance  . 137 


CONTENTS  xiii 

PAGE 

86.  Relation  of  motor  torque  curve  to  voltage  regulation  of  system 
— Regulation   coefficient   of  system — Stability   coefficient   of 
system 138 

87.  Effect  of  momentum  on  the  stability  of  the  motor — Regulation 

of  overload  capacity — Gradual  approach  to  instability  .    .    .    .    141 

CHAPTER  VII.     HIGHER  HARMONICS  IN  INDUCTION  MOTORS. 

88.  Component  torque  curves  due  to  the  higher  harmonics  of  the 
impressed  voltage  wave,  in  a  quarter-phase  induction  motor; 
their  synchronous  speed  and  their  direction,  and  the  resultant 
torque  curve 144 

89.  The  component  torque  curves  due  to  the  higher  harmonics  of 
the  impressed  voltage  wave,  in  a  three-phase  induction  motor — 
True  three-phase   and  six-phase  winding — Tl  e  single-phase 
torque  curve  of  the  third  harmonic 147 

90.  Component  torque  curves  of  normal  frequency,  but  higher 
number  of  poles,  due  to  the  harmonics  of  the  space  distribu- 
tion of  the  winding  in  the  air-gap  of  a  quarter-phase  motor — 
Their  direction  and  synchronous  speeds 150 

91.  The  same  in  a  three-phase  motor— Discussion  of  the  torque 
components  due  to  the  time  harmonics  of  higher  frequency 
and  normal  number  of  poles,  and  the  space  harmonics  of  normal 
frequency  and  higher  number  of  poles 154 

92.  Calculation  of  the  coefficients  of  the  trigonometric  series  repre- 
senting the  space  distribution  of  quarter-phase,  six-phase  and 
three-phase,  full  pitch  and  fractional  pitch  windings 155 

93.  Calculation  of  numerical  values  for  0,  ^,  Yz,  %  pitch  defi- 
ciency, up  to  the  21st  harmonic* 157 

CHAPTER  VII.     SYNCHRONIZING  INDUCTION  MOTORS. 

94.  Synchronizing  induction  motors  when  using  common  secondary 
resistance 159 

95.  Equation  of  motor  torque,  total  torque  and  synchronizing 
torque  of  two  induction  motors  with  common  secondary  rheo- 
stat       160 

96.  Discussion  of  equations — Stable  and  unstable  position — Maxi- 
mum synchronizing  power   at   45°  phase   angle — Numerical 
instance , 163 

CHAPTER  IX.     SYNCHRONOUS  INDUCTION  MOTOR. 

97.  Tendency  to  drop  into  synchronism,  of  single  circuit  induction 
motor  secondary — Motor  or  generator  action  at  synchronism — 
Motor  acting  as  periodically  varying  reactance,  that  is,  as 
reaction  machine — Low  power  factor — Pulsating  torque  below 
synchronism,  due  to  induction  motor  and  reaction  machine 
torque  superposition 166 


xiv  CONTENTS 

CHAPTER  X.     HYSTERESIS  MOTOR. 

PAGE 

98.  Rotation  of  iron  disc  in  rotating  magnetic  field — Equations — 
Motor  below,  generator  above  synchronism 168 

99.  Derivation  of  equations  from  hysteresis  law — Hysteresis  torque 

of  standard  induction  motor,  and  relation  to  size 169 

100.  General    discussion    of    hysteresis    motor — Hysteresis    loop 
collapsing  or  expanding 170 

CHAPTER  XI.     ROTARY  TERMINAL  SINGLE-PHASE  INDUCTION  MOTORS. 

101.  Performance   and   method    of   operation   of   rotary  terminal 
single-phase   induction  Motor — Relation   of   motor  speed   to 
brush  speed  and  slip  corresponding  to  the  load 172 

102.  Application  of  the  principle  to  a  self-starting  single-phase  power 
motor  with  high  starting  and  accelerating  torque,  by  auxiliary 
motor  carrying  brushes 173 

CHAPTER  XII.  FREQUENCY  CONVERTER  OR  GENERAL  ALTERNATING 
CURRENT  TRANSFORMER. 

103.  The  principle  of  the  frequency  converter  or  general  alternating 
current  transformer — Induction  motor  and  transformer  special 
cases — Simultaneous   transformation   between   primary   elec- 
trical and  secondary  electrical  power,  and  between  electrical 
and  mechanical  power — Transformation  of  voltage  and  of  fre- 
quency— The  air-gap  and  its  effect 176 

104.  Relation  of  e.m.f.,  frequency,  number  of  turns  and  exciting 
current 177 

105.  Derivation  of  the  general  alternating  current  transformer — 
Transformer  equations  and  induction  motor  equations,  special 
cases  thereof 178 

106.  Equation  of  power  of  general  alternating  current  transformer   .    182 

107.  Discussion:  between  synchronism  and  standstill — Backward 
driving — Beyond    synchronism — Relation    between    primary 
electrical,  secondary  electrical  and  mechanical  power   .    .    .    .184 

108.  Calculation  of  numerical  instance 185 

109.  The   characteristic    curves:  regulation    curve,    compounding 
curve — Connection  of  frequency  converter  with  synchronous 
machine,  and  compensation  for  lagging  current — Derivation  of 
equation  and  numerical  instance 186 

110.  Over-synchronous    operation — Two    applications,    as   double 
synchronous  generator,  and  as  induction  generator  with  low 
frequency  exciter 190 

111.  Use  as  frequency  converter — Use  of  synchronous  machine  or 
induction  machine  as  second  machine — Slip  of  frequency — 
Advantage  of  frequency  converter  over  motor  generator   .    .    .    191 

112.  Use  of  frequency  converter — Motor  converter,  its  advantages 
and  disadvantages — Concatenation  for  multispeed  operation   .    192 


CONTENTS  xv 

CHAPTER  XIII.     SYNCHRONOUS  INDUCTION  GENERATOR. 

PAGE 

113.  Induction  machine  as  asynchronous  motor  and  asynchronous 
generator 194 

114.  Excitation  of  induction  machine  by  constant  low  frequency 
voltage    in    secondary — Operation   below   synchronism,    and 
above  synchronism 195 

115.  Frequency  and  power  relation — Frequency  converter  and  syn- 
chronous induction  generator 196 

116.  Generation  of  two  different  frequencies,  by  stator  and  by  rotor .    198 

117.  Power  relation  of  the  two  frequencies — Equality  of  stator  and 
rotor  frequency:  double    synchronous  generator — Low  rotor 
frequency:  induction  generator  with   low  frequency  exciter, 
Stanley  induction  generator 198 

118.  Connection  of  rotor  to  stator  by  commutator — Relation  of  fre- 
quencies and  powers  to  ratio  of  number  of  turns  of  stator  and 
rotor 199 

119.  Double  synchronous  alternator — General  equation — Its  arma- 
ture reaction      201 

120.  Synchronous  induction  generator  with  low  frequency  excita- 
tion— (a)  Stator  and  rotor  fields  revolving  in  opposite  direc- 
tion— (6)  In  the  same  direction — Equations 203 

121.  Calculation  of  instance,  and  regulation  of  synchronous  induc- 
tion generator  with  oppositely  revolving  fields 204 

122.  Synchronous  induction  generator  with  stator  and  rotor  fields 
revolving  in  the  same  direction — Automatic  compounding  and 
over-compounding,  on  non-inductive  load — Effect  of  inductive 
load 205 

123.  Equations  of  synchronous  induction  generator  with  fields  re- 
volving in  the  same  direction 207 

124.  Calculation  of  numerical  instance     .    .    . 209 

CHAPTER  XIV.     PHASE  CONVERSION  AND  SINGLE-PHASE  GENERATION. 

125.  Conversion  between  single-phase  and  poly  phase  requires  energy 
•storage — Capacity,    inductance    and   momentum   for   energy 
storage — Their  size  and  cost  per  Kva   .    , 212 

126.  Industrial  importance  of  phase  conversion  from  single-phase  to 
polyphase,  and  from  balanced  polyphase  to  single-phase  .    .    .   213 

127.  Monocyclic  devices — Definition  of  monocyclic  as  a  system  of 
polyphase  voltages  with  essentially  single-phase  flow  of  energy 
— Relativity  of  the  term — The  monocyclic  triangle  for  single- 
phase  motor  starting 214 

128.  General  equations  of  the  monocyclic  square 216 

129.  Resistance — inductance    monocyclic    square — Numerical    in- 
stance on  inductive  and  on  non-inductive  load — Discussion   .   218 

130.  Induction  phase  converter — Reduction  of  the  device  to  the 
simplified  diagram  of  a  double  transformation 220 

131.  General  equation  of  the  induction  phase  converter 222 


xvi  CONTENTS 

PAGE 

132.  Numerical   instance — Inductive   load — Discussion   and   com- 
parisons with  monocyclic  square 223 

133.  Series  connection  of  induction  phase  converter  in  single-phase 
induction  motor  railway — Discussion  of  its  regulation  ....   226 

134.  Synchronous  phase  converter  and  single-phase  generation — 
Control  of  the  unbalancing  of  voltage  due  to  single-phase  load, 
by  stationary  induction  phase  balancing  with  reverse  rotation 

of  its  polyphase  system — Synchronous  phase  balancer.    .    .    .   227 

135.  Limitation  of  single-phase  generator  by  heating  of  armature 
coils — By  double  frequency  pulsation  of  armature  reaction — 
Use  of  squirrel  cage  winding  in  field — Its  size — Its  effect  on  the 
momentary  short  circuit  current 229 

136.  Limitation  of  the  phase  converter  in  distributing  single-phase 
load    into    a   balanced    polyphase   system — Solution    of   the 
problem  by  the  addition  of  a  synchronous  phase  balancer  to  the 
synchronous  phase  converter — Its  construction 230 

137.  The  various  methods  of  taking  care  of  large  single-phase  loads — 
Comparison  of  single-phase  generator  with  polyphase  generator 
and  phase  converter — Apparatus  economy 232 

CHAPTER  XV.     SYNCHRONOUS  RECTIFIERS. 

138.  Rectifiers  for  battery  charging — For  arc  lighting — The  arc  ma- 
chine as  rectifier — Rectifiers  for  compounding  alternators — 
For  starting  synchronous  motors — Rectifying  commutator — 
Differential   current    and   sparking   on   inductive   load — Re- 
sistance bipass — Application  to  alternator  and  synchronous 
motor 234 

139.  Open  circuit  and  short  circuit  rectification — Sparking  with 
open   circuit   rectification    on   inductive   load,    and   shift    of 
brushes 237 

140.  Short  circuit  rectification  on  non-inductive  and  on  inductive 
load,  and  shift  of  brushes — Rising  differential  current  and  flash- 
ing around  the  commutator — Stability  limit  of  brush  position, 
between  sparking  and  flashing — Commutating  e.m.f.  resulting 
from  unsymmetrical  short  circuit  voltage  at  brush  shift — 
Sparkless  rectification      239 

141.  Short  circuit  commutation  in  high  inductance,  open  circuit 
commutation  in  low  inductance  circuits — Use  of  double  brush 
to  vary  short  circuit — Effect  of  load — Thomson  Houston  arc 
machine — Brush  arc  machine — Storage  battery  charging   .    .   243 

142.  Reversing  and  contact  making  rectifier — Half  wave  rectifier 
and  its  disadvantage  by  unidirectional  magnetization  of  trans- 
former— The  two  connections  full  wave  contact  making  recti- 
fiers— Discussion  of  the  two  types  of  full  wave  rectifiers — 
The  mercury  arc  rectifier 245 

143.  Rectifier   with   intermediary   segments — Polyphase   rectifica- 
tion— Star  connected,  ring  connected  and  independent  phase 


CONTENTS  xvii 

PAGE 

rectifiers — Y  connected  three-phase  rectifier — Delta  connected 
three-phase  rectifier — Star  connected  quarter-phase  rectifier — 
Quarter-phase  rectifier  with  independent  phases — Ring  con- 
nected quarter-phase  rectifier — Wave  shapes  and  their  discus- 
sion— Six-phase  rectifier 250 

144.  Ring  connection  or  independent  phases  preferable  with  a  large 
number  of  phases — Thomson  Houston  arc  machine  as  con- 
stant current  alternator  with  three-phase  star  connected  rectifier 
— Brush   arc   machine   as   constant   current   alternator   with 
quarter-phase  rectifiers  in  series  connection 254 

145.  Counter  e.m.f.  shunt  at  gaps  of  polyphase  ring  connected 
rectifier — Derivation  of  counter  e.m.f.  from  synchronous  mo- 
tor— Leblanc's  Panchahuteur — Increase  of  rectifier  output  with 
increasing  number  of  phases 255 

146.  Discussion:  stationary  rectifying  commutator  with  revolving 
brushes — Permutator — Rectifier  with  revolving  transformer — 
Use   of  synchronous   motor  for  phase  splitting    in    feeding 
rectifying. commutator:  synchronous  converter — Conclusion   .   257 

CHAPTER  XVI.     REACTION  MACHINES. 

147.  Synchronous  machines  operating  without  field  excitation  .    .   260 

148.  Operation  of  synchronous  motor  without  field  excitation  de- 
pending on  phase  angle  between  resultant  m.m.f .  and  magnetic 
flux,  caused  by  polar  field  structure — Energy  component  of 
reactance 261 

149.  Magnetic  hysteresis  as  instance  giving  energy  component  of 
reactance,  as  effective  hysteretic  resistance 262 

150.  Make    and   break    of    magnetic    circuit — Types    of    reaction 
machines — Synchronous  induction  motor — Reaction  machine 

as  converter  from  d.-c.  to  a.-c 263 

151.  Wave  shape  distortion  in  reaction  machine,  due  to  variable 
reactance,  and  corresponding  hysteresis  cycles 264 

152.  Condition  of  generator  and  of  motor  action  of  the  reactance 
machine,  as  function  of  the  current  phase 267 

153.  Calculation  of  reaction  machine  equation — Power  factor  and 
maximum  power 268 

154.  Current,  power  and  power  factor — Numerical  instance     .    .    .   271 

155.  Discussion — Structural  similarity  with  inductor  machine   .    .   272 

CHAPTER  XVII.     INDUCTOR  MACHINES. 

156.  Description  of  inductor  machine  type — Induction  by  pulsating 
unidirectional  magnetic  flux 274 

157.  Advantages  and  disadvantages  of  inductor  type,  with  regards 

to  field  and  to  armature .  - 275 

158.  The  magnetic  circuit  of  the  inductor  machine,  calculation  of 
magnetic  flux  and  hysteresis  loss 276 


xviii  CONTENTS 

PAGE 

159.  The  Stanley  type  of  inductor  alternator — The  Alexanderson 
high  frequency  inductor  alternator  for  frequencies  of  100,000 
cycles  and  over 279 

160.  The  Eickemeyer  type  of  inductor  machine  with  bipolar  field — 
The  converter  from  direct  current  to  high  frequency  alternating 
current  of  the  inductor  type 280 

161.  Alternating  current  excitation  of  inductor  machine,  and  high 
frequency    generation    of   pulsating    amplitude.     Its    use    as 
amplifier — Amplification  of  telephone  currents  by  high  fre- 
quency inductor  in  radio  communication 281 

162.  Polyphase  excitation  of  inductor,   and  the  induction  motor 
inductor  frequency  converter 282 

163.  Inductor  machine  with  reversing  flux,  and  magneto  communi- 
cation— Transformer  potential  regulator  with  magnetic  com- 
mutation     284 

164.  The  interlocking  pole  type  of  field  design  in  alternators  and 
commutating  machines 286 

165.  Relation  of  inductor  machine  to  reaction  machine — Half  syn- 
chronous   operation    of    standard    synchronous    machine    as 
inductor  machine 287 

CHAPTER  XVIII.     SURGING  OF  SYNCHRONOUS  MOTORS. 

166.  Oscillatory  adjustment  of  synchronous  motor  to  changed  con- 
dition of  load — Decrement  of  oscillation — Cumulative  oscil- 
lation by  negative  decrement 288 

167.  Calculation  of  equation  of  electromechanical  resonance    .    .    .  289 

168.  Special  cases  and  example 292 

169.  Anti -surging  devices  and  pulsation  of  power 293 

170.  Cumulative  surging — Due  to  the  lag  of  some  effect  behind  its 
cause — Involving  a  frequency  transformation  of  power    .    .    .   296 

CHAPTER  XIX.    ALTERNATING  CURRENT  MOTORS  IN  GENERAL. 

171.  Types  of  alternating-current  motors 300 

172.  Equations  of  coil  revolving  in  an  alternating  field 302 

173.  General  equations  of  alternating-curreat  motor 304 

174.  Polyphase  induction  motor,  equations 307 

175.  Polyphase  induction  motor,  slip,  power,  torque 310 

176.  Polyphase  induction  motor,  characteristic  constants   ....   312 

177.  Polyphase  induction  motor,  example 313 

178.  Singlephase  induction  motor,  equations 314 

179.  Singlephase  induction  motor,  continued 316 

180.  Singlephase  induction  motor,  example 318 

181.  Polyphase  shunt  motor,  general    . 319 

182.  Polyphase  shunt  motor,  equations 320 

183.  Polyphase  shunt  motor,  adjustable  speed  motor 321 

184.  Polyphase  shunt  motor,  synchronous  speed  motor 323 

185.  Polyphase  shunt  motor,  phase  control  by  it 324 

186.  Polyphase  shunt  motor,  short-circuit  current  under  brushes  .   327 

187.  Polyphase  series  motor,  equations 327 

188.  Polyphase  series  motor,  example 330 


CONTENTS  xix 

CHAPTER  XX.     SINGLE-PHASE  COMMUTATOR  MOTORS. 

PAGE 

189.  General:  proportioning  of  parts  of  a.-c.  commutator  motor 
different  from  d.-c 331 

190.  Power  factor:  low  field  flux  and  high  armature  reaction  re- 
quired— Compensating  winding  necessary  to  reduce  armature 
self-induction 332 

191.  The  three  circuits  of  the  single-phase  commutator  motor — 
Compensation    and    over-compensation — Inductive   compen- 
sation— Possible  power  factors  .    .    t 336 

192.  Field    winding    and    compensating    winding:   massed    field 
winding  and  distributed  compensating  winding — Under-com- 
pensation  at  brushes,  due  to  incomplete  distribution  of  com- 
pensating winding 338 

193.  Fractional  pitch  armature  winding  to  secure  complete  local 
compensation — Thomson's  repulsion  motor — Eickemeyer  in- 
ductively compensated  series  motor 339 

194.  Types  of  varying  speed  single-phase  commutator  motors:  con- 
ductive and  inductive  compensation;  primary  and  secondary 
excitation;  series  and  repulsion  motors — Winter — Eichberg — 
Latour  motor — Motor  control  by  voltage  variation  and  by 
change  of  type .%......     341 

195.  The  quadrature  magnetic  flux  and  its  values  and  phases  in  the 
different  motor  types ...    ....   345 

196.  Commutation:  e.m.f.  of  rotation  and  e.m.f.  of  alternation — 
Polyphase  system  of  voltages — Effect  of  speed ".    .   347 

197.  Commutation  determined  by  value  and  phase  of  short  circuit 
current — High  brush  contact  resistance  and  narrow  brushes   .   349 

198.  Commutator  leads — Advantages  and  disadvantages  of  resist- 
ance leads  in  running  and  in  starting      351 

199.  Counter  e.m.f.  in  commutated  coil:  partial,  but  not  com- 
plete neutralization  possible 354 

200.  Commutating  field — Its  required  intensity  and  phase  rela- 
tions: quadrature  field 356 

201.  Local  commutating  pole — Neutralizing  component  and  revers- 
ing component  of  commutating  field — Discussion  of  motor 
types  regarding  commutation 358 

202.  Motor  characteristics :  calculation  of  motor — Equation  of  cur- 
rent, torque,  power  .    .    .-  .    ...    ,    .    ......    .    '.    .    .    .  361 

203.  Speed  curves  and  current  curves  of  motor — Numerical  instance 
— Hysteresis  loss  increases,   short  circuit  current  decreases 
power  factor 364 

204.  Increase  of  power  factor  by  lagging  field   magnetism,   by 
resistance  shunt  across  field 366 

205.  Compensation  for  phase  displacement  and  control  of  power 
factor  by  alternating  current  commutator  motor  with  lagging 
field  flux,  as  effective  capacity — Its  use  in  induction  motors  and 
other  apparatus ...'.>.".    ...    .    .   370 

206.  Efficiency  and  losses:  the  two  kinds  of  core  loss  .    .    .    .    ,    .  370 


xx  CONTENTS 

PAGE 

207.  Discussion  of  motor  types:  compensated  series  motors:  con- 
ductive and  inductive  compensation — Their  relative  advan- 
tages and  disadvantages   371 

208.  Repulsion  motors:  lagging  quadrature  flux — Not  adapted  to 
speeds  much  above  synchronism — Combination   type:  series 
repulsion  motor 373 

209.  Constructive  differences — Possibility  of  changing  from  type  to 
type,  with  change  of  speed  or  load 375 

210.  Other  commutator  motors:  shunt    motor — Adjustable  speed 
polyphase    induction    motor — Power    factor    compensation: 
Heyland  motor — Winter-Eichberg  motor 377 

211.  Most  general  form  of  single-phase  commutator  motor,  with  two 
stator  and  two  rotor  circuits  and  two  brush  short  circuits   .    .381 

212.  General  equation  of  motor 382 

213.  Their  application  to  the  different  types  of  single-phase  motor 
with  series  characteristic .   383 

214.  Repulsion  motor:  Equations 385 

215.  Continued 388 

216.  Discussion  of  commutation  current  and  commutation  factor   .   391 

217.  Repulsion  motor  and  repulsion  generator    .    .    ....    .    .-.    .   394 

218.  Numerical  instance .    ...    .    .    -    .    .    .    .   395 

219.  Series  repulsion  motor:  equations    .    ...    1  "...    .    .    .    .   397 

220.  Continued . 398 

221.  Study  of  commutation — Short  circuit  current  under  brushes   .   403 

222.  Commutation  current  .    ......  /  .    ,    .    .    .  V  .    .    .    .   404 

223.  Effect  of  voltage  ratio  and  phase,  on  commutation    .    ..  „    .    .   406 

224.  Condition  of  vanishing  commutation  current      .    .    ...    .    .   408 

225.  Numerical  example .".....    .    .    .    .    .   411 

226.  Comparison  of  repulsion  motor  and  various  series  repulsion 
motor      ..••.. 414 

227.  Further  example — Commutation  factors     >    .    .    .    ...    •    •   415 

228.  Over-compensation — Equations ,    .    .    .   418 

229.  Limitation  of  preceding  discussion — Effect  and  importance  of 
transient  in  short  circuit  current   .    .    .    ,    .    .  . .    ...    .    .    .    .   419 

CHAPTER  XXI.     REGULATING  POLE  CONVERTER. 

230.  Change  of  converter  ratio  by  changing  position  angle  between 
brushes  and  magnetic  flux,  and  by  change  of  wave  shape   .    .   422 
A.  Variable  ratio  by  change  of  position  angle  between  com- 
mutator brushes  and  resultant  magnetic  flux 422 

231.  Decrease  of  a.-c.  voltage  by  shifting  the  brushes — By  shifting 
the  magnetic  flux — Electrical  shifting  of  the  magnetic  flux  by 
varying  the  excitation  of  the  several  sections  of  the  field  pole  .  422 

232.  Armature  reaction  and  commutation — Calculation  of  the  re- 
sultant armature  reaction  of  the  converter  with  shifted  mag- 
netic flux 426 

233.  The  two  directions  of  shift  flux,  the  one  spoiling,  the  other 


CONTENTS  xxi 

PAGE 
improving  commutation — Demagnetizing   armature   reaction 

and  need  of  compounding  by  series  field 429 

B.  Variable  ratio  by  change  of  the  wave  shape  of  the  Y  voltage  429 

234.  Increase  and  decrease  of  d.-c.  voltage  by  increase  or  decrease 
of  maximum  a.-c.  voltage  by  higher  harmonic — Illustration 

by  third  and  fifth  harmonic • 430 

235.  Use  of  the  third  harmonic  in  the  three-phase  system — Trans- 
former connection  required  to  limit  it  to  the  local  converter 
circuit — Calculation  of  converter  wave  as  function  of  the  pole 
arc 432 

236.  Calculation    of    converter    wave    resulting   from    reversal   of 
middle  of  pole  arc 435 

237.  Discussion      436 

238.  Armature    reaction    and    commutation — Proportionality    of 
resultant  armature  reaction  to  deviation  of  voltage  ratio  from 
normal 437 

239.  Commutating  flux  of  armature  reaction  of  high  a.-c.  voltage — 
Combination  of  both  converter  types,  the  wave  shape  distor- 
tion for  raising,  the  flux  shift  for  lowering  the  a.-c.  voltage — 
Use  of  two  pole  section,  the  main  pole  and  the  regulating  pole   .   437 

240.  Heating  and  rating — Relation   of  currents   and  voltages  in 
standard  converter 439 

241.  Calculation  of  the  voltages  and  currents  in  the  regulating  pole 
converter 440 

242.  Calculating  of  differential  current,  and  of  relative  heating  of 
armature  coil .    .    .    .    .   442 

243.  Average  armature  heating  of  n  phase  converter .   444 

244.  Armature  heating  and  rating  of  three-phase  and  of  six-phase 
regulating  pole  converter 445 

245.  Calculation  of  phase  angle  giving  minimum  heating  or  maxi- 
mum rating 446 

246.  Discussion  of  conditions  giving  minimum  heating — Design — 
Numerical  instance 448 

CHAPTER  XXII.     UNIPOLAR  MACHINES. 

Homopolar  Machines — Acyclic  Machines 

247.  Principle  of  unipolar,   homopolar  or  acyclic  machine — The 
problem  of  high  speed  current  collection— Fallacy  of  unipolar 
induction  in  stationary  conductor — Immaterial  whether  mag- 
net standstill  or  revolves — The  conception  of  lines  of  magnetic 
force 450 

248.  Impossibility  of  the  coil  wound  unipolar  machine — All  electro- 
magnetic induction  in  turn  must  be  alternating — Illustration 

of  unipolar  induction  by  motion  on  circular  track 452 

249.  Discussion  of  unipolar  machine  design — Drum  type  and  disc 
type — Auxiliary    air-gap — Double    structure — Series    connec- 
tion of  conductors  with  separate  pairs  of  collector  rings    .    .    .   454 


xxii  CONTENTS 

PAGE 

250.  Unipolar  machine  adapted  for  low  voltage,  or  for  large  size  high 
speed  machines — Theoretical  absence  of  core  loss — Possibility 
of  large  core  loss,  by  eddies,  in  core  and  in  collector  rings,  by 
pulsating  armature  reaction 456 

251.  Circular    magnetization    produced    by    armature    reaction — 
Liability  to  magnetic  saturation  and  poor  voltage  regulation — 
Compensating  winding — Most  serious  problem  the  high  speed 
collector  rings 457 

252.  Description  of  unipolar  motor  meter 458 

CHAPTER  XXIII.     REVIEW. 

253.  Alphabetical   list    of    machines:    name,    definition,    principal 
characteristics,  advantages  and  disadvantages 459 

CHAPTER  XXIV.     CONCLUSION. 

254.  Little  used  and  unused  types  of  apparatus — Their  knowledge 
important  due  to  the  possibility  of  becoming  of  great  industrial 
importance — Illustration  by  commutating  pole  machine   .    .   472 

255.  Change  of  industrial  condition  may  make  new  machine  types 
important — Example    of   induction    generator   for    collecting 
numerous  small  water  powers 473 

256.  Relative  importance  of  standard  types  and  of  special  types  of 
machines 474 

257.  Classiiication  of  machine  types  into  induction,  synchronous, 
commutating  and  unipolar  machines — Machine  belonging  to 
two  and  even  three  types 474 

INDEX  .   477 


THEORY  AND  CALCULATION  OF 
ELECTRICAL  APPARATUS 


CHAPTER  I 

SPEED  CONTROL  OF  INDUCTION  MOTORS 
I.  STARTING  AND  ACCELERATION 

1.  Speed  control  of  induction  motors  deals  with  two  problems : 
to  produce  a  high. torque  over  a  wide  range  of  speed  down  to 
standstill,  for  starting  and-  acceleration;  and  to  produce  an 
approximately  constant  speed  for  a  wide  range  of  load,  for 
constant-speed  operation. 

In  its  characteristics,  the  induction  motor  is  a  shunt  motor, 
that  is,  it  runs  at  approximately  constant  speed  for  all  loads, 
and  this  speed  is  synchronism  at  no-load.  At  speeds  below  full 
speed,  and  at  standstill,  the  torque  of  the  motor  is  low  and  the 
current  high,  that  is,  the  starting-torque  efficiency  and  especially 
the  apparent  starting-torque  efficiency  are  low. 

Where  starting  with  considerable  load,  and  without  excessive 
current,  is  necessary,  the  induction  motor  thus  requires  the  use 
of  a  resistance  in  the  armature  or  secondary,  just  as  the  direct- 
current  shunt  motor,  and  this  resistance  must  be  a  rheostat, 
that  is,  variable,  so  as  to  have  maximum  resistance  in  starting, 
and  gradually,  or  at  least  in  a  number  of  successive  steps,  cut 
out  the  resistance  during  acceleration. 

This,  however,  requires  a  wound  secondary,  and  the  squirrel- 
cage  type  of  rotor,  which  is  the  simplest,  most  reliable  and  there- 
fore most  generally  used,  is  not  adapted  for  the  use  of  a  start- 
ing rheostat.  With  the  squirrel-cage  type  of  induction  motor, 
starting  thus  is  usually  done — and  always  with  large  motors — 
by  lowering  the  impressed  voltage  by  autotransformer,  often 
in  a  number  of  successive  steps.  This  reduces  the  starting 
current,  but  correspondingly  reduces  the  starting  torque,  as  it 
does  not  change  the  apparent  starting-torque  efficiency. 

The  higher  the  rotor  resistance,  the  greater  is  the  starting 
torque,  and  the  less,  therefore,  the  starting  current  required  for 

1 


2  ELECTRICAL  APPARATUS 

a  given  torque  when  starting  by  autotransformer.  However, 
high  rotor  resistance  means  lower  efficiency  and  poorer  speed 
regulation,  and  this  limits  the  economically  permissible  resistance 
in  the  rotor  or  secondary. 

Discussion  of  the  starting  of  the  induction  motor  by  arma- 
ture rheostat,  and  of  the  various  speed-torque  curves  produced 
by  various  values  of  starting  resistance  in  the  induction-motor 
secondary,  are  given  in  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena"  and  in  " Theoretical  Elements  of  Electrical 
Engineering." 

As  seen,  in  the  induction  motor,  the  (effective)  secondary  re- 
sistance should  be  as  low  as  possible  at  full  speed,  but  should 
be  high  at  standstill — very  high  compared  to  the  full-speed 
value — and  gradually  decrease  during  acceleration,  to  maintain 
constant  high  torque  from  standstill  to  speed.  To  avoid  the 
inconvenience  and  complication  of  operating  a  starting  rheostat, 
various  devices  have  been  proposed  and  to  some  extent  used,  to 
produce  a  resistance,  which  automatically  increases  with  in- 
creasing slip,  and  thus  is  low  at  full  speed,  and  higher  at  standstill. 

A.  Temperature  Starting  Device 

2:  A  resistance  material  of  high  positive  temperature  coeffi- 
cient of  resistance,  such  as  iron  and  other  pure  metals,  operated 
at  high  temperature,  gives  this  effect  to  a  considerable  extent: 
with  increasing  slip,  that  is,  decreasing  speed  of  the  motor,  the 
secondary  current  increases.  If  the  dimensions  of  the  secondary 
resistance  are  chosen  so  that  it  rises  considerably  in  tempera- 
ture, by  the  increase  of  secondary  current,  the  temperature  and 
therewith  the  resistance  increases. 

Approximately,  the  temperature  rise,  and  thus  the  resistance 
rise  of  the  secondary  resistance,  may  be  considered  as  propor- 
tional to  the  square  of  the  secondary-current,  ii,  that  is,  repre- 
sented by: 

r  =  r°  (1  +  on2).  (1) 

As  illustration,  consider  a  typical  induction  motor,  of  the 
constants : 

e0  =  110; 

y0  =  g  -  jb  =  0.01  -  0.1  j; 
ZQ  =  r0+  jx<>  =  0.1  +  0.3J; 
Zl  =  n+jxi  =  0.1  +  0.3J; 

the  speed-torque  curve  of  this  motor  is  shown  as  A  in  Fig.  1. 


SPEED  CONTROL  3 

Suppose  now  a  resistance,  r,  is  inserted  in  series  into  the  sec- 
ondary circuit,  which  when  cold — that  is,  at  light-load — equals 
the  internal  secondary  resistance: 

r°  =  n  =  0.1, 

but  increases  so  as  to  double  with  100  amp.  passing  through  it. 
This  resistance  can  then  be  represented  by: 

r  =  ro  (i  +  ^2  10-4) 

=    0.1    (1    +    I?   10-4), 


INDUCTION   MOTOR 

Z,  =  r,-l-.3j 

SPEED  CONTROL  BY  POSITIVE  TEMPERATURE  COEFFICIENT  r, 
SPEED  CURVES 


FIG.  1. — High-starting  and  acceleration  torque  of  induction  motor  by  posi- 
tive temperature  coefficient  of  secondary  resistance. 


and  the  total  secondary  resistance  of  the  motor  then  is : 

r'i  =  ri  +  r0  (1  +  a  *\2) 
=  0.2  (1  +  0.5  iS  10-4). 


(2) 


To  calculate  the  motor  characteristics  for  this  varying  resist- 
ance, r'i,  we  use  the  feature,  that  a  change  of  the  secondary  re- 
sistance of  the  induction  motor  changes  the  slip,  s,  in  proportion 
to  the  change  of  resistance,  but  leaves  the  torque,  current,  power- 
factor,  torque  efficiency,  etc.,  unchanged,  as  shown  on  page 
322  of  "  Theoretical  Elements  of  Electrical  Engineering."  We 
thus  calculate  the  motor  for  constant  secondary  resistance,  n, 
but  otherwise  the  same  constants,  in  the  manner  discussed  on 
page  318  of  "  Theoretical  Elements  of  Electrical  Engineering. " 


4  ELECTRICAL  APPARATUS 

This  gives  curve  A  of  Fig.  1.     At  any  value  of  torque,  T,  corre- 
sponding to  slip,  s,  the  secondary  current  is: 


i  =  e  A«i2  +  «22> 

herefrom  follows  by  (2)  the  value  of  r\,  and  from  this  the  new 
value  of  slip: 

s'  -=-  s  =  r'i  -i-  n.  (3) 

The  torque,  T,  then  is  plotted  against  the  value  of  slip,  s',  and 
gives  curve  B  of  Fig.  1.  As  seen,  5  gives  practically  constant 
torque  over  the  entire  range  from  near  full  speed,  to  standstill. 

Curve  B  has  twice  the  slip  at  load,  as  A,  as  its  resistance  has 
been  doubled. 

3.  Assuming,  now,  that  the  internal  resistance,  TI,  were  made 
as  low  as  possible,  r\  =  0.05,  and  the  rest  added  as  external 
resistance  of  high  temperature  coefficient:  r°  =  0.05,  giving  the 
total  resistance  : 

r'iN=  0.1  (1  +  0.5  ^2  10-4).  (4) 

This  gives  the  same  resistance  as  curve  A  :  r\  =  0.1,  at  light- 
load,  where  it  is  small  and  the  external  part  of  the  resistance  cold. 
But  with  increasing  load  the  resistance,  r'i,  increases,  and  the 
motor  gives  the  curve  shown  as  C  in  Fig.  1. 

As  seen,  curve  C  is  the  same  near  synchronism  as  A,  but  in 
starting  gives  twice  as  much  torque  as  A,  due  to  the  increased 
resistance, 

C  and  A  thus  are  directly  comparable:  both  have  the  same 
constants  and  same  speed  regulation  and  other  performance  at 
speed,  but  C  gives  much  higher  torque  at  standstill  and  during 
acceleration. 

For  comparison,  curve  A'  has  been  plotted  with  constant 
resistance  rt  =  0.2,  so  as  to  compare  with  B. 

Instead  of  inserting  an  external  resistance,  it  would  be  pref- 
erable to  use  the  internal  resistance  of  the  squirrel  cage,  to  in- 
crease in  value  by  temperature  rise,  and  thereby  improve  the 
starting  torque. 

Considering  in  this  respect  the  motor  shown  as  curve  C.  At 
standstill,  it  is:  ii  =  153;  thus  r\  =  0.217;  while  cold,  the  re- 
sistance is:  r'i  =  0.1.  This  represents  a  resistance  rise  of  117 
per  cent.  At  a  temperature  coefficient  of  the  resistance  of  0.35, 
this  represents  a  maximum  temperature  rise  of  335°C.  As  seen, 


SPEED  CONTROL  5 

by  going  to  temperature  of  about  350°C.  in  the  rotor  conductors 
—which  naturally  would  require  fireproof  construction — it  be- 
comes possible  to  convert  curve  A  into  C,  or  A'  into  B,  in  Fig.  1. 
Probably,  the  high  temperature  would  be  permissible  only  in 
the  end  connections,  or  the  squirrel-cage  end  ring,  but  then,  iron 
could  be  used  as  resistance  material,  which  has  a  materially 
higher  temperature  coefficient,  and  the  required  temperature 
rise  thus  would  probably  be  no  higher. 

B.  Hysteresis  Starting  Device 

4.  Instead  of  increasing  the  secondary  resistance  with  increas- 
ing slip,  to  get  high  torque  at  low  speeds,  the  same  result  can  be 
produced  by  the  use  of  an  effective  resistance,  such  as  the  effect- 
ive or  equivalent  resistance  of  hysteresis,  or  of  eddy  currents. 

As  the  frequency  of  the  secondary  current  varies,  a  magnetic 
circuit  energized  by  the  secondary  current  operates  at  the  varying 
frequency  of  the  slip,  s. 

At  a  given  current,  i\,  the  voltage  required  to  send  the  current 
through  the  magnetic  circuit  is  proportional  to  the  frequency, 
that  is,  to  s.  Hence,  the  susceptance  is  inverse  proportional 
to  s: 

V;-\.  ;   •:          (5) 

The  angle  of  hysteretic  advance* of  phase,  a,  and  the  power- 
factor,  in  a  closed  magnetic  circuit,  are  independent  of  the 
frequency,  and  vary  relatively  little  with  the  magnetic  density 
and  thus  the  current,  over  a  wide  range,1  thus  may  approxi- 
mately be  assumed  as  constant.  That  is,  the  hysteretic  con- 
ductance is  proportional  to  the  susceptance: 

g'  =  b'  tan  a.  (6) 

Thus,  the  exciting  admittance,  of  a  closed  magnetic  circuit 
of  negligible  resistance  and  negligible  eddy-current  losses,  at  the 
frequency  of  slip,  s,  is  given  by: 

Y'  =  g'  -  jb'  =  b'  (tan  a  -  j) 

=  l-jb  =  b(t^a-j)  (7) 

O  CO 

1  "Theory  and  Calculation  of  Alternating-current  Phenomena," 
Chapter  XII. 


6  ELECTRICAL  APPARATUS 

Assuming  tan  a  =  0.6,  which  is  a  fair  value  for  a  closed  mag- 
netic circuit  of  high  hysteresis  loss,  it  is: 


the  exciting  admittance  at  slip,  s. 

Assume  then,  that  such  an  admittance,  F',  is  connected  in  series 
into  the  secondary  circuit  of  the  induction  motor,  for  the  pur- 
pose of  using  the  effective  resistance  of  hysteresis,  which  in- 
creases with  the  frequency,  to  control  the  motor  torque  curve. 

The  total  secondary  impedance  then  is: 


-("+*!)  +*(*>+£)•  <«> 

where  :  Y  =  g  —  jb  is  the  admittance  of  the  magnetic  circuit  at 
full  frequency,  and 

y  =  Vg*  +  &2. 

5.  For  illustration,  assume  that  in  the  induction  motor  of  the 
constants  : 

eQ  =  100; 
Fo  =  0.02  -  0.2  j; 
Z0  =  0.05  +  0.15  j; 
Zi  =  0.05  +  0.15J; 

a  closed  magnetic  circuit  is  connected  into  the  secondary,  of  full 
frequency  admittance, 

Y  =  g-  jb', 
and  assume: 

g  =  0.66; 

6  =  4; 
thus,  by  (8)  : 

Z\  =  (0.05  +  0.11  s)  +  0.335  js.  (9) 

The  characteristic  curves  of  this  induction  motor  with  hysteresis 
starting  device  can  now  be  calculated  in  the  usual  manner,  dif- 
fering from  the  standard  motor  only  in  that  Zi  is  not  constant, 
and  the  proper  value  of  riy  x^  and  m  has  to  be  used  for  every 
slip,  s. 

Fig.  2  gives  the  speed-torque  curve,  and  Fig.  3  the  load  curves 
of  this  motor. 


SPEED  CONTROL  7 

For  comparison  is  shown,  as  T7',  in  dotted  lines,  the  torque 
curve  of  the  motor  of  constant  secondary  resistance,  and  of  the 
constants : 

Fo  =  0.01  -  0.1 .7; 
Z0  =  0.01  +  0.3  j; 
Z,  =  0.1  +  0.3J; 

As  seen,  the  hysteresis  starting  device  gives  higher  torque  at 
standstill  and  low  speeds,  with  less  slip  at  full  speed,  thus  a 
materially  superior  torque  curve. 


INDUCTION  MOTOR 
Y0=.02-.2j;   Z 0=.05+. 15 j;     e0= 

,QQ5js 

SPEED  CONTROL  BY  HYSTERESIS 
SPEED  CURVES 


FIG.  2. — Speed  curves  of  induction  motor  with  hysteresis  starting  device. 

p  represents  the  power-factor,  rj  the  efficiency,  7  the  apparent 
efficiency,  V  the  torque  efficiency  and  7'  the  apparent  torque 
efficiency. 

However,  T  corresponds  to  a  motor  of  twice  the  admittance 
and  half  the  impedance  of  T'.  That  is,  to  get  approximately 
the  same  output,  with  the  hysteresis  device  inserted,  as  without 
it,  requires  a  rewinding  of  the  motor  for  higher  magnetic  density, 
the  same  as  would  be  produced  in  T'  by  increasing  the  voltage 
\/2  times. 

It  is  interesting  to  note  in  comparing  Fig.  2  with  Fig.  1,  that 
the  change  in  the  torque  curve  at  low  and  medium  speed,  pro- 
duced by  the  hysteresis  starting  device,  is  very  similar  to  that 
•produced  by  temperature  rise  of  the  secondary  resistance;  at 


8 


ELECTRICAL  APPARATUS 


speed,  however,  the  hysteresis  device  reduces  the  slip,  while  the 
temperature  device  leaves  it  unchanged. 

The  foremost  disadvantage  of  the  use  of  the  hysteresis  device 
is  the  impairment  of  the  power-factor,  as  seen  in  Fig.  3  as  p. 

The  introduction  of  the  effective  resistance  representing  the 
hysteresis  of  necessity  introduces  a  reactance,  which  is  higher 
than  the  resistance,  and  thereby  impairs  the  motor  characteristics. 

Comparing  Fig.  3  with  Fig.   176,  page  319  of  "  Theoretical 


INDUCTION   MOTOR 


=.05-1-  .15  j,    6  o=100 


(.05  +  .11  s)  +  .335  js 

SPEED  CONTROL  BY  HYSTERESIS 

SPEED  CURVES 


0  15202530354045505560657075 


FIG.  3. — Load  curves  of  induction  motor  with  hysteresis  starting  device. 

Elements  of  Electrical  Engineering,"  which  gives  the  load  curves 
of  Tr  of  Fig.  2,  it  is  seen  that  the  hysteresis  starting  device  reduced 
the  maximum  power-factor,  p,  from  91  per  cent,  to  84  per  cent., 
and  the  apparent  efficiency,  7,  correspondingly. 
This  seriously  limits  the  usefulness  of  the  device. 

C.  Eddy-current  Starting  Device 

6.  Assuming  that,  instead  of  using  a  well-laminated  magnetic 
circuit,  and  utilizing  hysteresis  to  give  the  increase  of  effective 
resistance  with  increasing  slip,  we  use  a  magnetic  circuit  having 
very  high  eddy-current  losses:  very  thick  laminations  or  solid 
iron,  or  we  directly  provide  a  closed  high-resistance  secondary 
winding  around  the  magnetic  circuit,  which  is  inserted  into  the 
induction-motor  secondary  for  increasing  the  starting  torque. 


SPEED  CONTROL  9 

The  susceptance  of  the  magnetic  circuit  obviously  follows  the 
same  law  as  when  there  are  no  eddy  currents.  That  is : 

b'=bg  (10) 

At  a  given  current,  ii,  energizing  the  magnetic  circuit,  the  in- 
duced voltage,  and  thus  also  the  voltage  producing  the  eddy 
currents,  is  proportional  to  the  frequency.  The  currents  are 
proportional  to  the  voltage,  and  the  eddy-current  losses,  there- 
fore, are  proportional  to  the  square  of  the  voltage.  The  eddy- 
current  conductance,  g,  thus  is  independent  of  the  frequency. 

The  admittance  of  a  magnetic  circuit  consuming  energy  by 
eddy  currents  (and  other  secondary  currents  in  permanent  closed 
circuits),  of  negligible  hysteresis  loss,  thus  is  represented,  as 
function  of  the  slip,  by  the  expression: 

Y'-g-j--  (11) 

o 

Connecting  such  an  admittance  in  series  to  the  induction- 
motor  secondary,  gives  the  total  secondary  impedance:  • 

7'         7  l 

"Y> 

/                         0         \     |       •    /  &  \  Y10\ 

=    I  ~1>2  I          3   I  SXl / — P\~V  ^         ' 

Assuming : 

g  -  b.  (13) 

That  is,  45°  phase  angle  of  the  exciting  circuit  of  the  magnetic 
circuit  at  full  frequency — which  corresponds  to  complete  screen- 
ing of  the  center  of  the  magnet  core — we  get : 


*  +  - 


Fig.  4  shows  the  speed  curves,  and  Fig.  5  the  load  curves, 
calculated  in  the  standard  manner,  of  a  motor  with  eddy-current 
starting  device  in  the  secondary,  of  the  constants: 

eo  =  100; 
Fo  =  0.03  -  0.3  j; 

Z0  =  0.033  +  0.1  j; 
Zi  =  0.033  +  0.1J; 

b  =  3; 


10 
thus: 


ELECTRICAL  APPARATUS 


7.  As  seen,  the  torque  curve  has  a  very  curious  shape:  a 
maximum  at  7  per  cent,  slip,  and  a  second  higher  maximum  at 
standstill. 

The  torque  efficiency  is  very  high  at  all  speeds,  and  prac- 
tically constant  at  82  per  cent,  from  standstill  to  fairly  close  of 
full  speed,  when  it  increases. 


INDUCTION   MOTOR 
Y0=03-.3j;    Z0=.033+  1j;  e0=1OO 


SPEED  CONTROL  BY  EDDIES 
SPEED  CURVES 


FIG.  4. — Speed  curves  of  induction  motor  with  eddy-current  starting  device. 

But  the  power-factor  is  very  poor,  reaching  a  maximum  of 
78  per  cent,  only,  and  to  get  the  output  from  the  motor,  required 
rewinding  it  to  give  the  equivalent  of  a  \/3  times  as  high  voltage. 

For  comparison,  in  dotted  lines  as  Tr  is  shown  the  torque  curves 
of  the  standard  motor,  of  same  maximum  torque.  As  seen,  in 
the  motor  with  eddy-current  starting  device,  the  slip  at  load  is 
very  small,  that  is,  the  speed  regulation  very  good.  Aside  from 
the  poor  power-factor,  the  motor  constants  would  be  very 
satisfactory. 

The  low  power-factor  seriously  limits  the  usefulness  of  the 
device. 

By  differently  proportioning  the  eddy-current  device  to  the 
secondary  circuit,  obviously  the  torque  curve  can  be  modified 


SPEED  CONTROL 


11 


and  the  starting  torque  reduced,  the  depression  in  the  torque 
curve  between  full-speed  torque  and  starting  torque  eliminated, 
etc. 

Instead  of  using  an  external  magnetic  circuit,  the  magnetic 
circuit  of  the  rotor  or  induction-motor  secondary  may  be  used, 
and  in  this  case,  instead  of  relying  on  eddy  currents,  a  definite 
secondary  circuit  could  be  utilized,  in  the  form  of  a  second 
squirrel  cage  embedded  deeply  in  the  rotor  iron,  that  is,  a  double 
squirrel-cage  motor. 


0.5  iJ 


INDUCTION   MOTOR 
Y0=.03-.3j  ;    Z0  =  .033-»-  .1j  ; 


SPEED  CONTROL  BY  EDDIES 
LOAD  CURVES 


0  1 


5  2 


0  25  30  35  4,0  4J5  50  5.5  606. 


\ 


5  7.0  7,5 


150 

140 

130 

120 

110 

100 

90 

80 

70 

60 

50 

40 

80 

20 

10 

0 


FIG.  5. — Load  curves  of  induction  motor  with  eddy-current  starting  device. 

In  the  discussion  of  the  multiple  squirrel-cage  induction  motor, 
Chapter  II,  we  shall  see  speed-torque  curves  of  the  character  as 
shown  in  Fig.  4.  By  the  use  of  the  rotor  iron  as  magnetic  cir- 
cuit, the  impairment  of  the  power-factor  is  somewhat  reduced, 
so  that  the  multiple  squirrel-cage  motor  becomes  industrially 
important. 

A  further  way  of  utilizing  eddy  currents  for  increasing  the 
effective  resistance  at  low  speeds,  is  by  the  use  of  deep  rotor 
bars.  By  building  the  rotor  with  narrow  and  deep  slots  filled 
with  solid  deep  bars,  eddy  currents  in  these  bars  occur  at  higher 
frequencies,  or  unequal  current  distribution.  That  is,  the  cur- 
rent flows  practically  all  through  the  top  of  the  bars  at  the  high 


12  ELECTRICAL  APPARATUS 

frequency  of  low  motor  speeds,  thus  meeting  with  a  high  resist- 
ance. With  increasing  motor  speed  and  thus  decreasing 
secondary  frequency,  the  current  penetrates  deeper  into  the  bar, 
until  at  full  speed  it  passes  practically  uniformly  throughout 
the  entire  bar,  in  a  circuit  of  low  resistance — but  somewhat 
increased  reactance. 

The  deep-bar  construction,  the  eddy-current  starting  device 
and  the  double  squirrel-cage  construction  thus  are  very  similar 
in  the  motor-performance  curves,  and  the  double  squirrel  cage, 
which  usually  is  the  most  economical  arrangement,  thus  will  be 
discussed  more  fully  in  Chapter  II. 

II.  CONSTANT-SPEED  OPERATION 

8.  The  standard  induction  motor  is  essentially  a  constant-speed 
motor,  that  is,  its  speed  is  practically  constant  for  all  loads, 
decreasing  slightly  with  increasing  load,  from  synchronism  at 
no-load.  It  thus  has  the  same  speed  characteristics  as  the  direct- 
current  shunt  motor,  and  in  principle  is  a  shunt  motor. 

In  the  direct-current  shunt  motor,  the  speed  may  be  changed 
by:  resistance  in  the  armature,  resistance  in  the  field,  change  of 
the  voltage  supply  to  the  armature  by  a  multivolt  supply  circuit, 
as  a  three-wire  system,  etc. 

In  the  induction  motor,  the  speed  can  be  reduced  by  inserting 
resistance  into  the  armature  or  secondary,  just  as  in  the  direct- 
current  shunt  motor,  and  involving  the  same  disadvantages: 
the  reduction  of  speed  by  armature  resistance  takes  place  at  a 
sacrifice  of  efficiency,  and  at  the  lower  speed  produced  by  arma- 
ture resistance,  the  power  input  is  the  same  as  it  would  be  with 
the  same  motor  torque  at  full  speed,  while  the  power  output  is 
reduced  by  the  reduced  speed.  That  is,  speed  reduction  by 
armature  resistance  lowers  the  efficiency  in  proportion  to  the 
lowering  of  speed.  The  foremost  disadvantage  of  speed  control 
by  armature  resistance  is,  however,  that  the  motor  ceases  to  be 
a  constant-speed  motor,  and  the  speed  varies  with  the  load: 
with  a  given  value  of  armature  resistance,  if  the  load  and  with  it 
the  armature  current  drops  to  one-half,  the  speed  reduction  of 
the  motor,  from  full  speed,  also  decreases  to  one-half,  that  is, 
the  motor  speeds  up,  and  if  the  load  comes  off,  the  motor  runs 
up  to  practically  full  speed.  Inversely,  if  the  load  increases,  the 
speed  slows  down  proportional  to  the  load. 

With  considerable  resistance  in  the  armature,  the  induction 


SPEED  CONTROL  13 

motor  thus  has  rather  series  characteristic  than  shunt  character- 
istic, except  that  its  speed  is  limited  by  synchronism. 

Series  resistance  in  the  armature  thus  is  not  suitable  to  produce 
steady  running  at  low  speeds. 

To  a  considerable  extent,  this  disadvantage  of  inconstancy  of 
speed  can  be  overcome: 

(a)  By  the  use  of  capacity  or  effective  capacity  in  the  motor 
secondary,  which  contracts  the  range  of  torque  into  that  of 
approximate  resonance  of  the  capacity  with  the  motor  inductance, 
and  thereby  gives  fairly  constant  speed,  independent  of  the  load, 
at  various  speed  values  determined  by  the  value  of  the  capacity. 

(6)  By  the  use  of  a  resistance  of  very  high  negative  tempera- 
ture coefficient  in  the  armature,  so  that  with  increase  of  load  and 
current  the  resistance  decreases  by  its  increase  of  temperature, 
and  thus  keeps  approximately  constant  speed  over  a  wide  range 
of  load. 

Neither  of  these  methods,  however,  avoids  the  loss  of  efficiency 
incident  to  the  decrease  of  speed. 

9.  There  is  no  method  of  speed  variation  of  the  induction 
motor  analogous  to  field  control  of  the  shunt  motor,  or  change 
of  the  armature  supply  voltage  by  a  multivolt  supply  system. 
The  field  excitation  of  the  induction  motor  is  by  what  may  be 
called  armature  reaction.  That  is,  the  same  voltage,  impressed 
upon  the  motor  primary,  gives  the  energy  current  and  the  field 
exciting  current,  and  the  field  excitation  thus  can  not  be  varied 
without  varying  the  energy  supply  voltage,  and  inversely. 
Furthermore,  the  no-load  speed  of  the  induction  motor  does  not 
depend  on  voltage  or  field  strength,  but  is  determined  by 
synchronism. 

The  speed  of  the  induction  motor  can,  however,  be  changed: 

(a)  By  changing  the  impressed  frequency,  or  the  effective 
frequency. 

(6)  By  changing  the  number  of  poles  of  the  motor. 

Neither  of  these  two  methods  has  any  analogy  in  the  direct- 
current  shunt  motor:  the  direct-current  shunt  motor  has  no  fre- 
quency relation  to  speed,  and  its  speed  is  not  determined  by  the 
number  of  poles,  nor  is  it  feasible,  with  the  usual  construction 
of  direct-current  motors,  to  easily  change  the  number  of  poles. 

In  the  induction  motor,  a  change  of  impressed  frequency  corre- 
spondingly changes  the  synchronous  speed.  The  effect  of  a 
change  of  frequency  is  brought  about  by  concatenation  of  the 


14  ELECTRICAL  APPARATUS 

motor  with  a  second  motor,  or  by  internal  concatenation  of  the 
motor:  hereby  the  effective  frequency,  which  determines  the 
no-load  or  synchronous  speed,  becomes  the  difference  between 
primary  and  secondary  frequency. 

Concatenation  of  induction  motors  is  more  fully  discussed  in 
Chapter  III. 

As  the  no-load  or  synchronous  speed  of  the  induction  motor 
depends  on  the  number  of  poles,  a  change  of  the  number  of  poles 
changes  the  motor  speed.  Thus,  if  in  a  60-cycle  induction  motor, 
the  number  of  poles  is  changed  from  four  to  six  and  to  eight,  the 
speed  is  changed  from  1800  to  1200  and  to  900  revolutions  per 
minute. 

This  method  of  speed  variation  of  the  induction  motor,  by 
changing  the  number  of  poles,  is  the  most  convenient,  and  such 
"multispeed  motors"  are  extensively  used  industrially. 

A.  Pyro-electric  Speed  Control 

10.  Speed  control  by  resistance  in  the  armature  or  secondary 
has  the  disadvantage  that  the  speed  is  not  constant,  but  at 
a  change  of  load  and  thus  of  current,  the  voltage  consumed 
by  the  armature  resistance,  and  therefore  the  speed  changes. 
To  give  constancy  of  speed  over  a  range  of  load  would  require 
a  resistance,  which  consumes  the  same  or  approximately  the 
same  voltage  at  all  values  of  current.  A  resistance  of  very 
high  negative  temperature  coefficient  does  this :  with  increase  of 
current  and  thus  increase  of  temperature,  the  resistance  decreases, 
and  if  the  decrease  of  resistance  is  as  large  as  the  increase  of 
current,  the  voltage  consumed  by  the  resistance,  and  therefore 
the  motor  speed,  remains  constant. 

Some  pyro-electric  conductors  (see  Chapter  I,  of  "  Theory 
and  Calculation  of  Electric  Circuits")  have  negative  tempera- 
ture coefficients  sufficiently  high  for  this  purpose.  Fig.  6  shows 
the  current-resistance  characteristic  of  a  pyro-electric  conductor, 
consisting  of  cast  silicon  (the  same  of  which  the  characteristic 
is  given  as  rod  II  in  Fig.  6  of  "  Theory  and  Calculation  of  Electric 
Circuits").  Inserting  this  resistance,  half  of  it  and  one  and  one- 
half  of  it  into  the  secondary  of  the  induction  motor  of  constants : 
eo  =  110;  Fo  =  0.01  -  0.1j;Zo  =0.1  +  0.3  j;  Zl  =  0.1  +  0.3  j 
gives  the  speed-torque  curves  shown  in  Fig.  7. 

The  calculation  of  these  curves  is  as  follows:  The  speed- 
torque  curve  of  the  motor  with  short-circuited  secondary,  r  =  0, 


SPEED  CONTROL 


15 


RESISTANCE  OF 
PYRO  ELECTRIC  CONDUCTOR 
[SILICONRODNO.il.    FIG. 6 
"ELECTRIC  CIRCUITS" 


FIG.  6. — Variation  of  resistance  of  pyro-electric  conductor,  with  current. 


PYRO-ELECTRIC  RESISTANCE  IN  SECONDARY  OF  INDUCTION  MOTOR,  e0=110 

Y0=.01-.1j  ;  Z0=.1+.3j  ;  Z,-.1  +  .3j  ;  r  =  2,4.6 

SPEED  CONTROL  BY  PYRO  ELECTRIC  CONDUCTOR. 

SPEED  CURVES. 


FIG.  7. — Speed  control  of  induction  motor  by  pyro-electric  conductor, 

speed  curves 


16  ELECTRICAL  APPARATUS 

is  calculated  in  the  usual  way  as  described  on  page  318  of 
"  Theoretical  Elements  of  Electrical  Engineering."  For  any 
value  of  slip,  s,  and  corresponding  value  of  torque,  T,  the  secondary 
current  is  i\  —  e  V«i2  -p«22-  To  this  secondary  current  corre- 
sponds, by  Fig.  6,  the  resistance,  r,  of  the  pyro-electric  conductor, 
and  the  insertion  of  r  thus  increases  the  slip  in  proportion  to  the 

increased  secondary  resistance:   -       —  >  where  r\  =  0.1  in  the 

?*! 

present  instance.  This  gives,  as  corresponding  to  the  torque, 
T}  the  slip: 


where  s  =  slip  at  torque,  T,  with  short-circuited  armature,  or 
resistance,  r\. 

As  seen  from  Fig.  7,  very  close  constant-speed  regulation  is 
produced  by  the  use  of  the  pyro-electric  resistance,  over  a  wide 
range  of  load,  and  only  at  light-load  the  motor  speeds  up. 

Thus,  good  constant-speed  regulation  at  any  speed  below 
synchronism,  down  to  very  low  speeds,  would  be  produced  — 
at  a  corresponding  sacrifice  of  efficiency,  however  —  by  the  use 
of  suitable  pyro-electric  conductors  in  the  motor  armature. 

The  only  objection  to  the  use  of  such  pyro-electric  resistances 
is  the  difficulty  of  producing  stable  pyro-electric  conductors,  and 
permanent  terminal  connections  on  such  conductors. 

B.  Condenser  Speed  Control 

11.  The  reactance  of  a  condenser  is  inverse  proportional  to 
the  frequency,  that  of  an  inductance  is  directly  proportional  to 
the  frequency.  In  the  secondary  of  the  induction  motor,  the 
frequency  varies  from  zero  at  synchronism,  to  full  frequency  at 
standstill.  If,  therefore,  a  suitable  capacity  is  inserted  into  the 
secondary  of  an  induction  motor,  there  is  a  definite  speed,  at 
which  inductive  reactance  and  capacity  reactance  are  equal  and 
opposite,  that  is,  balance,  and  at  and  near  this  speed,  a  large 
current  is  taken  by  the  motor  and  thus  large  torque  developed, 
while  at  speeds  considerably  above  or  below  this  resonance  speed, 
the  current  and  thus  torque  of  the  motor  are  small. 

The  use  of  a  capacity,  or  an  effective  capacity  (as  polariza- 
tion cell  or  aluminum  cell)  in  the  induction-motor  secondary 
should  therefore  afford,  at  least  theoretically,  a  means  of  speed 
control  by  varying  the  capacity. 


SPEED  CONTROL  17 

Let,  in  an  induction  motor  : 

YO  =  g  —  jb  =  primary  exciting  admittance; 

Z0  =  rQ  +  jx0  =  primary  self-inductive  impedance; 

Zi  =  n  +  jxi  =  internal  self-inductive  impedance,  at  full 

frequency; 

and  let  the  condenser,  C,  be  inserted  into  the  secondary  circuit. 
The  capacity  reactance  of  C  is 


k 
at  full  frequency,  and  -  at  the  frequency  of  slip,  s. 

s 

The  total  secondary  impedance,  at  slip,  s,  thus  is  : 


=  ••!+/(<«,-  (2) 


and  the  secondary  current: 

T  sE  .  se 


=  E  (a-!  -  ja2), 
where  : 


«2    = 


m 
m  =  ri2  4 


(4) 


The  further  calculation  of  the  condenser  motor,  then,  is  the 
same  as  that  of  the  standard  motor.1 
12.  Neglecting  the  exciting  current: 

/oo  =  EY 
the  primary  current  equals  the  secondary  current: 

/•r/i 

and  the  primary  impressed  voltage  thus  is: 


1  "  Theoretical  Elements  of  Electrical  Engineering,"  4th  edition,  p.  318. 
2 


18  ELECTRICAL  APPARATUS 

and,  substituting  (3)  and  rearranging,  gives: 


.  £•  (5) 

Oi  +  sr0)  +  j  («Si  +  sx0  -  -J 

or,  absolute: 


(6) 

(ri  +  sr0)2  +  (sxi  +  sx 


The  torque  of  the  motor  is : 

T   =   62«! 

and,  substituting  (4)  and  (6) : 

sri602 


+  sr0)2+  '"-     '    -        <l/x  (7) 


As  seen,  this  torque  is  a  maximum  in  the  range  of  slip,  s, 
where  the  second  term  in  the  denominator  vanishes,  while  for 
values  of  s,  materially  differing  therefrom,  the  second  term  in  the 
denominator  is  large,  and  the  torque  thus  small. 

That  is,  the  motor  regulates  for  approximately  constant  speed 
near  the  value  of  s,  given  by: 


-{-  SXQ  --  =  0, 


that  is  : 


So  =       --  (8) 

X0  +  Xi 


and  SQ  =  1,   that  is,   the  motor  gives  maximum   torque  near 
standstill,  for: 

k  =  z0  4-  XL  (9) 

13.  As  instances  are  shown,  in  Fig.  8,  the  speed-torque  curves 
of  a  motor  of  the  constants  : 

7o  =  0.01  -  O.lj, 

ZQ  =  Zl  =  0.1  +  0.3  j, 


SPEED  CONTROL 


19 


for  the  values  of  capacity  reactance: 

k  =  0,  0.012,  0.048,  0.096,  0.192,  0.3,  0.6— denoted  respectively 

by  1,  2,  3,  4,  5,  6,  7. 

The  impressed  voltage  of  the  motor  is  assumed  to  be  varied 
with  the  change  of  capacity,  so  as  to  give  the  same  maximum 
torque  for  all  values  of  capacity. 

The  volt-ampere  capacity  of  the  condenser  is  given,  at  the 
frequency  of  slip,  s,  by: 


substituting  (3)  and  (6),  this  gives: 

«?/?„ 

Q'  = 


(n 


k\2 
+  sxo J 


SPEEDCONTROLOF  INDUCTION  MOTOR  BY  CONDENSER   IN  SECONDARY 

Y0=.01-.1j;    Z0=.1+.3j;    Z.=  . 

CAPACITY  REACTANCE   k  :  (1):  0;  (2)   .012;  (3)   .048; 
(4)  .096;  (5)  .192;  (6)  .3;  (7)  .6 


FIG.  8. — Speed  control  of  induction  motor  by  condenser  in  secondary  circuit. 

Speed  curves. 

and,  compared  with  (7),  this  gives: 

Q'  =  7  T- 

At  full  frequency,  with  the  same  voltage  impressed  upon  the 
condenser,  its  volt-ampere  capacity,  and  thus  its  60-cycle  rating, 
would  be: 

Cl  ^l  K      rn 

o  =  --=--  i> 


20  ELECTRICAL  APPARATUS 

As  seen,  a  very  large  amount  of  capacity  is  required  for  speed 
control.  This  limits  its  economic  usefulness,  and  makes  the 
use  of  a  cheaper  form  of  effective  or  equivalent  capacity  desirable. 

C.  Multispeed  Motors 

14.  The  change  of  speed  by  changing  the  number  of  poles,  in 
the  multispeed  induction  motor,  involves  the  use  of  fractional- 
pitch  windings:  a  primary  turn,  which  is  of  full  pole  pitch  for 
a  given  number  of  motor  poles,  is  fractional  pitch  for  a  smaller 
number  of  poles,  and  more  than  full  pitch  for  a  larger  number 
of  poles.  The  same  then  applies  to  the  rotor  or  secondary,  if 
containing  a  definite  winding.  The  usual  and  most  frequently 
employed  squirrel-cage  secondary  obviously  has  no  definite 
number  of  poles,  and  thus  is  equally  adapted  to  any  number  of 
poles. 

As  an  illustration  may  be  considered  a  three-speed  motor 
changing  between  four,  six  and  eight  poles. 

Assuming  that  the  primary  winding  is  full-pitch  for  the  six- 
polar  motor,  that  is,  each  primary  turn  covers  one-sixth  of  the 
motor  circumference.  Then,  for  the  four-polar  motor,  the 
primary  winding  is  %  pitch,  for  the  eight-polar  motor  it  is  ££ 
pitch — which  latter  is  effectively  the  same  as  %  pitch. 

Suppose  now  the  primary  winding  is  arranged  and  connected 
as  a  six-polar  three-phase  winding.  Comparing  it  with  the 
same  primary  turns,  arranged  as  a  four-polar  three-phase  wind- 
ing, or  eight-polar  three-phase  winding,  the  turns  of  each  phase 
can  be  grouped  in  six  sections : 

Those  which  remain  in  the  same  phase  when  changing  to  a 
winding  for  different  number  of  poles. 

Those  which  remain  in  the  same  phase,  but  are  reversed  when 
changing  the  number  of  poles. 

Those  which  have  to  be  transferred  to  the  second  phase. 

Those  which  have  to  be  transferred  to  the  second  phase  in  the 
reverse  direction. 

Those  which  have  to  be  transferred  to  the  third  phase. 

Those  which  have  to  be  transferred  to  the  third  phase  in  the 
reverse  direction. 

The  problem  of  multispeed  motor  design  then  is,  so  to  arrange 
the  windings,  that  the  change  of  connection  of  the  six  coil  groups 
of  each  phase,  in  changing  from  one  number  of  poles  to  another, 
is  accomplished  with  the  least  number  of  switches. 


SPEED  CONTROL  21 

15.  Considering  now  the  change  of  motor  constants  when 
changing  speed  by  changing  the  number  of  poles.  Assuming 
that  at  all  speeds,  the  same  primary  turns  are  connected  in  series, 
and  are  merely  grouped  differently,  it  follows,  that  the  self- 
inductive  impedances  remain  essentially  unchanged  by  a  change 
of  the  number  of  poles  from  n  to  n'  .  That  is: 


With  the  same  supply  voltage  impressed  upon  the  same  number 
of  series  turns,  the  magnetic  flux  per  pole  remains  unchanged 
by  the  change  of  the  number  of  poles.  The  flux  density,  there- 
fore, changes  proportional  to  the  number  of  poles: 

&       rf 
B    :=  n  ' 

therefore,  the  ampere-turns  per  pole  required  for  producing  the 
magnetic  flux,  also  must  be  proportional  to  the  number  of  poles: 


However,  with  the  same  total  number  of  turns,  the  number  of 
turns  per  pole  are  inverse  proportional  to  the  number  of  poles  : 


N    ~  n' 

In  consequence  hereof,  the  exciting  currents,  at  the  same 
impressed  voltage,  are  proportional  to  the  square  of  the  number 
of  poles: 


and  thus  the  exciting  susceptances  are  proportional  to  the  square 
of  the  number  of  poles  : 


n 


The  magnetic  flux  per  pole  remains  the  same,  and  thus  the 
magnetic-flux  density,  and  with  it  the  hysteresis  loss  in  the 
primary  core,  remain  the  same,  at  a  change  of  the  number  of 
poles.  The  tooth  density,  however,  increases  with  increasing 
number  of  poles,  as  the  number  of  teeth,  which  carry  the  same 
flux  per  pole,  decreases  inverse  proportional  to  the  number  of 


22  ELECTRICAL  APPARATUS 

poles.  Since  the  tooth  densities  must  be  chosen  sufficiently  low 
not  to  reach  saturation  at  the  highest  number  of  poles,  and  their 
core  loss  is  usually  small  compared  with  that  in  the  primary  core 
itself,  it  can  be  assumed  approximately,  that  the  core  loss  of 
the  motor  is  the  same,  at  the  same  impressed  voltage,  regardless 
of  the  number  of  poles.  This  means,  that  the  exciting  con- 
ductance, g,  does  not  change  with  the  number  of  poles. 

Thus,  if  in  a  motor  of  n  poles,  we  change  to  nr  poles,  or  by  the 
ratio 

n' 

a  =  — , 
n 

the  motor  constants  change,  approximately: 
from:  to: 

Z0  =  r0  -f  JXQ,  Z0  =  r0  +  JXQ. 

Zi  =  rl  +  jxi,  Zi  =  TI  +  jxi. 

^o  =  g  -  jb,  Fo  =  g  -  jazb. 

16.  However,  when  changing  the  number  of  poles,  the  pitch 
of  the  winding  changes,  and  allowance  has  to  be  made  herefore 
in  the  constants:  a  fractional-pitch  winding,  due  to  the  partial 
neutralization  of  the  turns,  obviously  has  a  somewhat  higher 
exciting  admittance,  and  lower  self-inductive  impedance,  than 
a  full-pitch  winding. 

As  seen,  in  a  multispeed  motor,  the  motor  constants  at  the 
higher  number  of  poles  and  thus  the  lower  speed,  must  be 
materially  inferior  than  at  the  higher  speed,  due  to  the  increase 
of  the  exciting  susceptance,  and  the  performance  of  the  motor, 
and  especially  its  power-factor  and  thus  the  apparent  efficiency, 
are  inferior  at  the  lower  speeds. 

When  retaining  series  connection  of  all  turns  for  all  speeds, 
and  using  the  same  impressed  voltage,  torque  in  synchronous 
watts,  and  power  are  essentially  the  same  at  all  speeds,  that  is, 
are  decreased  for  the  lower  speed  and  larger  number  of  poles 
only  as  far  as  due  to  the  higher  exciting  admittance.  The  actual 
torque  thus  would  be  higher  for  the  lower  speeds,  and  approxi- 
mately inverse  proportional  to  the  speed. 

As  a  rule,  no  more  torque  is  required  at  low  speed  than  at 
high  speed,  and  the  usual  requirement  would  be,  that  the  multi- 
speed  motor  should  carry  the  same  torque  at  all  its  running 
speeds,  that  is,  give  a  power  proportional  to  the  speed. 

This  would  be  accomplished  by  lowering  the  impressed  voltage 


SPEED  CONTROL 


23 


for  the  larger  number  of  poles,  about  inverse  proportional  to  the 
square  root  of  the  number  of  poles : 


since  the  output  is  proportional  to  the  square  of  the  voltage. 

The  same  is  accomplished  by  changing  connection  from  multiple 
connection  at  higher  speeds  to  series  connection  at  lower  speeds, 
or  from  delta  connection  at  higher  speeds,  to  Y  at  lower  speeds. 

If,  then,  the  voltage  per  turn  is  chosen  so  as  to  make  the  actual 
torque  proportional  to  the  synchronous  torque  at  all  speeds,  that 


MULTISPEED  INDUCTION   MOTOR 
4  POLES  1800  REV. 


FIG.  9. — Load  curves  for  multispeed  induction  motor,  highest  speed,  four 

poles. 

is,  approximately  equal,  then  the  magnetic  flux  per  pole  and  the 
density  in  the  primary  core  decreases  with  increasing  number  of 
poles,  while  that  in  the  teeth  increases,  but  less  than  at  constant 
impressed  voltage. 

The  change  of  constants,  by  changing  the  number  of  poles  by 
the  ratio: 


thus  is : 
from: 


n' 

—  =  a 


€Q}    YQ,  ZQ,  Z\  to  60, 


ttZ 


and  the  characteristic  constant  is  changed  from  $  to  a2#. 

17.  As  numerical  instance  may  be  considered  a  60-cycle  100- 
volt  motor,  of  the  constants : 


24 


ELECTRICAL  APPARATUS 


MULTISPEED  INDUCTION   MOTOR 


ULTISPEEDINDUCTION  MOTOR 


FIG.  10. — Load  curvefof  multi- 
speed  induction  motor,  middle 
speed,  six  poles. 


FIG.  11. — Load  cuives  of  multi- 
speed  induction  motor,  low  speed, 
eight  poles. 


FIG.  1 

REV. 

1800 

MULTISPEED  INDUCTION   MOTOR 
4-6-8-POLES.  1800-1200-900  REV. 

1700 



•"—— 

—     —  . 

-»-.. 



•—--  — 

Si 

ifion 

•-•«» 

•—•  - 



—  , 

150(1 

1400 
1300 
1200 

1100 
1000 





• 

S2 

—  —  ... 



—  -—  . 

t 

MRS. 

"• 

% 

100 

900 

S3 

-.. 

-—  — 

^1 

90 

-800 
_700. 
_600 
.500 
-400 
-300 

.200 
100 

~z 

£ 

*^^ 
^ 

:^a= 

^2 

^      * 

=^=^ 

=5=" 

a^  •• 

S= 

•   '  — 

=^i-: 

—  •  ~-« 

P2 
•=-i= 

—  —  ~» 
1  —  -^ 

~~~- 

rl 

J2 

=-80 

III 

/ 

Pf 

s' 
^ 

^~ 

****' 

^3 

/' 

^X 

^ 

X 

\. 

60 

'/ 

/ 

y 

II. 

/ 

/ 

,s 

50 

'/,< 

/  / 

^' 

^ 

I  2^ 

,^x 

r^ 

fs 

40 

'// 

.^ 

^ 

^^ 

**" 
^' 

,^ 

^la 

30 

J 

<s 

s* 

^  —  • 

^ 

^" 

•**"* 

20 

•^y~* 

f*^Z. 

~*~ 

*f^~" 

10 

o"o 

5   1 

0   1 

5   2 

0   2 

5   3 

0  3 

5   4 

0   4 

5   5 

0   5 

5  6 

0   6 

5   7 

08Y^ 

.KW. 

2.  —  Comparison  of  load  curves  of  three-speed  induction  motor. 

SPEED  CONTROL 


25 


Four  poles,  1800  rev.:  Z0  =  r0  +  jx0  =  0.1  +  0.3.;; 

Zi  =  rl+jx1  =  0.1  +  0.3  j;  Y0  =  g  -  jb  =  0.01  -  0.05  j. 
Six  poles,  1200  rev.:  ZQ  =  r0  +  jz0  =  0.15  +  0.45  j] 
Zl  =  n+jxi  =  0.15  +  0.45  j;  Y0  =  g-jb  =  0.0067  -  0.0667J. 
Eight  poles,  900  rev. :  ZQ  =  r0  +  jxQ  =  0.2  +  0.6  j; 

Zj  =  n  +  jzi  =  0.2  +  0.6  j;  YQ  =  g  -  jb  =  0.005  -  0.1  j. 

Figs.  9,  10  and  11  show  the  load  curves  of  the  motor,  at  the 
three  different  speeds.     Fig.  12  shows  the  load  curves  once  more, 


MULTISPEED  INDUCTION   MOTOR 
4-6-8  POLES       1800-1200-900    REV 


110 


100 


'JO 


_7__7 


6-  -60 


_20 


.10 


100   200  300  400  500  600   700  800   900100011001200130014001500160017001800 

FIG.  13. — Speed  torque  curves  of  three-speed  induction  motor. 

with  all  three  motors  plotted  on  the  same  sheet,  but  with  the 
torque  in  synchronous  watts  (referred  to  full  speed  or  four- 
polar  synchronism)  as  abscissae,  to  give  a  better  comparison. 
S  denotes  the  speed,  I  the  current,  p  the  power-factor  and  7  the 
apparent  efficiency.  Obviously,  carrying  the  same  load,  that 
is,  giving  the  same  torque  at  lower  speed,  represents  less  power 
output,  and  in  a  multispeed  motor  the  maximum  power  output 
should  be  approximately  proportional  to  the  speed,  to  operate 
at  all  speeds  at  the  same  part  of  the  motor  characteristic.  There- 
fore, a  comparison  of  the  different  speed  curves  by  the  power 
output  does  not  show  the  performance  as  well  as  a  comparison 
on  the  basis  of  torque,  as  given  in  Fig.  12. 


26  ELECTRICAL  APPARATUS 

As  seen  from  Fig.  12,  at  the  high  speed,  the  motor  performance 
is  excellent,  but  at  the  lowest  speed,  power-factor  and  apparent 
efficiency  are  already  low,  especially  at  light-load. 

The  three  current  curves  cross :  at  the  lowest  speed,  the  motor 
takes  most  current  at  no-load,  as  the  exciting  current  is  highest ; 
at  higher  values  of  torque,  obviously  the  current  is  greatest  at 
the  highest  speed,  where  the  torque  represents  most  power. 

The  speed  regulation  is  equally  good  at  all  speeds. 

Fig.  13  then  shows  the  speed  curves,  with  revolutions  per 
minute  as  abscissae,  for  the  three  numbers  of  poles.  It  gives 
current,  torque  and  power  as  ordinates,  and  shows  that  the 
maximum  torque  is  nearly  the  same  at  all  three  speeds,  while 
current  and  power  drop  off  with  decrease  of  speed. 


CHAPTER  II 
MULTIPLE    SQUIRREL-CAGE    INDUCTION    MOTOR 

18.  In  an  induction  motor,  a  high-resistance  low-reactance 
secondary  is  produced  by  the  use  of  an  external  non-inductive 
resistance  in  the  secondary,  or  in  a  motor  with  squirrel-cage 
secondary,  by  small  bars  of  high-resistance  material  located  close 
to  the  periphery  of  the  rotor.  Such  a  motor  has  a  great  slip  of 
speed  under  load,  therefore  poor  efficiency  and  poor  speed  regu- 
lation, but  it  has  a  high  starting  torque  and  torque  at  low  and 
intermediate  speed.  With  a  low  resistance  fairly  high-reactance 
secondary,  the  slip  of  speed  under  load  is  small,  therefore  effi- 
ciency and  speed  regulation  good,  but  the  starting  torque  and 
torque  at  low  and  intermediate  speeds  is  low,  and  the  current 
in  starting  and  at  low  speed  is  large.  To  combine  good  start- 
ing with  good  running  characteristics,  a  non-inductive  resistance 
is  used  in  the  secondary,  which  is  cut  out  during  acceleration. 
This,  however,  involves  a  complication,  which  is  undesirable 
in  many  cases,  such  as  in  ship  propulsion,  etc.  By  arranging 
then  two  squirrel  cages,  one  high-resistance  low-reactance  one, 
consisting  of  high-resistance  bars  close  to  the  rotor  surface, 
and  one  of  low-resistance  bars,  located  deeper  in  the  armature 
iron,  that  is,  inside  of  the  first  squirrel  cage,  and  thus  of  higher 
reactance,  a  " double  squirrel-cage  induction  motor"  is  derived, 
which  to  some  extent  combines  the  characteristics  of  the  high- 
resistance  and  the  low-resistance  secondary.  That  is,  at  start- 
ing and  low  speed,  the  frequency  of  the  magnetic  flux  in  the  arma- 
ture, and  therefore  the  voltage  induced  in  the  secondary  winding 
is  high,  and  the  high-resistance  squirrel  cage  thus  carries  con- 
siderable current,  gives  good  torque  and  torque  efficiency,  while 
the  low-resistance  squirrel  cage  is  ineffective,  due  to  its  high 
reactance  at  the  high  armature  frequency.  At  speeds  near 
synchronism,  the  secondary  frequency,  being  that  of  slip,  is  low, 
and  the  secondary  induced  voltage  correspondingly  low.  The 
high-resistance  squirrel  cage  thus  carries  little  current  and  gives 
little  torque.  In  the  low-resistance  squirrel  cage,  due  to  its  low 
reactance  at  the  low  frequency  of  slip,  in  spite  of  the  relatively 

27 


28  ELECTRICAL  APPARATUS 

low  induced  e.m.f.,  considerable  current  is  produced,  which  is 
effective  in  producing  torque.  Such  double  squirrel-cage  induc- 
tion motor  thus  gives  a  torque  curve,  which  to  some  extent  is  a 
superposition  of  the  torque  curve  of  the  high-resistance  and  that 
of  the  low-resistance  squirrel  cage,  has  two  maxima,  one  at  low 
speed,  and  another  near  synchronism,  therefore  gives  a  fairly 
good  torque  and  torque  efficiency  over  the  entire  speed  range 
from  standstill  to  full  speed,  that  is,  combines  the  good  features 
of  both  types.  Where  a  very  high  starting  torque  requires 
locating  the  first  torque  maximum  near  standstill,  and  large  size 
and  high  efficiency  brings  the  second  torque  maximum  very  close 
to  synchronism,  the  drop  of  torque  between  the  two  maxima 
may  be  considerable.  This  is  still  more  the  case,  when  the  motor 
is  required  to  reverse  at  full  speed  and  full  power,  that  is;  a  very 
high  torque  is  required  at  full  speed  backward,  or  at  or  near 
slip  s  =  2.  In  this  case,  a  triple  squirrel  cage  may  be  used,  that 
is,  three  squirrel  cages  inside  of  each  other:  the  outermost,  of 
high  resistance  and  low  reactance,  gives  maximum  torque  below 
standstill,  at  backward  rotation;  the  second  squirrel  cage,  of 
medium  resistance  and  medium  reactance,  gives  its  maximum 
torque  at  moderate  speed;  and  the  innermost  squirrel  cage,  of 
low  resistance  and  high  reactance,  gives  its  torque  at  full  speed, 
near  synchronism. 

Mechanically,  the  rotor  iron  may  be  slotted  down  to  the  inner- 
most squirrel  cage,  so  as  to  avoid  the  excessive  reactance  of  a 
closed  magnetic  circuit,  that  is,  have  the  magnetic  leakage  flux 
or  self-inductive  flux  pass  an  air  gap. 

19.  In  the  calculation  of  the  standard  induction  motor,  it  is 
usual  to  start  with  the  mutual  magnetic  flux,  $,  or  rather  with 
the  voltage  induced  by  this  flux,  the  mutual  inductive  voltage 
E  =  e,  as  it  is  most  convenient,  with  the  mutual  inductive 
voltage,  e,  as  starting  point,  to  pass  to  the  secondary  current  by 
the  self-inductive  impedance,  to  the  primary  current  and  primary 
impressed  voltage  by  the  primary  self-inductive  impedance  and 
exciting  admittance. 

In  the  calculation  of  multiple  squirrel-cage  induction  motors, 
it  is  preferable  to  introduce  the  true  induced  voltage,  that  is, 
the  voltage  induced  by  the  resultant  magnetic  flux  interlinked 
with  the  various  circuits,  which  is  the  resultant  of  the  mutual 
and  the  self-inductive  magnetic  flux  of  the  respective  circuit. 
This  permits  starting  with  the  innermost  squirrel  cage,  and 


INDUCTION  MOTOR  29 

gradually  building  up  to  the  primary  circuit.     The  advantage 
hereof  is,  that  the  current  in  every  secondary  circuit  is  in  phase 

with  the  true  induced  voltage  of  this  circuit,  and  is  i\  =  —  > 

T\ 

where  ri  is  the  resistance  of  the  circuit.  As  e\  is  the  voltage 
induced  by  the  resultant  of  the  mutual  magnetic  flux  coming 
from  the  primary  winding,  and  the  self-inductive  flux  corre- 
sponding to  the  i\x\  of  the  secondary,  the  reactance,  x\,  does  not 
enter  any  more  in  the  equation  of  the  current,  and  e\  is  the 
voltage  due  to  the  magnetic  flux  which  passes  beyond  the  cir- 
cuit in  which  e^  is  induced.  In  the  usual  induction-motor  theory, 
the  mutual  magnetic  flux,  <£,  induces  a  voltage,  E,  which  produces 
a  current,  and  this  current  produces  a  self-inductive  flux,  $'i, 
giving  rise  to  a  counter  e.m.f.  of  self-induction  I\x^  which  sub- 
tracts from  E.  However,  the  self -inductive  flux,  $>'i,  interlinks 
with  the  same  conductors,  with  which  the  mutual  flux,  $>,  inter- 
links, and  the  actual  or  resultant  flux  interlinkage  thus  is  $1  = 
3>  —  3>'i,  and  this  produces  the  true  induced  voltage  e\  =  E  - 
IjXij  from  which  the  multiple  squirrel-cage  calculation  starts.1 

Double  Squirrel-cage  Induction  Motor 
20.  Let,  in  a  double  squirrel-cage  induction  motor : 

E2  =  true  induced  voltage  in  inner  squirrel  cage,  reduced 

to  full  frequency, 
/2  =  current,  and 
Z2  =  r2  +  jxz  =  self -inductive  impedance  at  full  frequency, 

reduced  to  the  primary  circuit. 
EI  =  true  induced  voltage  in  outer  squirrel  cage,  reduced 

to  full  frequency, 
/i  =  current,  and 
Zi  =  ri  +  jxi  =  self-inductive  impedance  at  full  frequency, 

reduced  to  primary  circuit. 
E  —  voltage  induced  in  secondary  and  primary  circuits  by 

mutual  magnetic  flux, 
EQ  —  voltage  impressed  upon  primary, 
/o  =  primary  current, 

Z0  =  TQ  -f  jxQ  =   primary  self -inductive  impedance,   and 
Yo  =  g  —  jb  =  primary  exciting  admittance. 

1  See    "Electric    Circuits",     Chapter    XII.     Reactance    of    Induction 
Apparatus. 


30  ELECTRICAL  APPARATUS 

The  leakage  reactance,  xz,  of  the  inner  squirrel  cage  is  that  due 
to  the  flux  produced  by  £he  current  in  the  inner  squirrel  cage, 
which  passes  between  the  two  squirrel  cages,  and  does  not  in- 
clude the  reactance  due  to  the  flux  resulting  from  the  current, 
72,  which  passes  beyond  the  outer  squirrel  cage,  as  the  latter  is 
mutual  reactance  between  the  two  squirrel  cages,  and  thus  meets 
the  reactance,  x\. 

It  is  then,  at  slip  s: 

'•-£ 

'•-£  » 

/o  =  /2  +  1 1  +  Y0E.  (3) 

and: 

X2  72.  (4) 


#<,  =  E  +  ZQ/O.  (6) 

The  leakage  flux  of  the  outer  squirrel  cage  is  produced  by  the 
m.m.f.  of  the  currents  of  both  squirrel  cages,  /i  +  72,  and  the 
reactance  voltage  of  this  squirrel  cage,  in  (5),  thus  isjxi  (7i  -f  /2). 

As  seen,  the  difference  between  EI  and  E2  is  the  voltage  in- 
duced by  the  flux  which  leaks  between  the  two  squirrel  cages,  in 
the  path  of  the  reactance,  x2,  or  the  reactance  voltage,  x2/2;  the 
difference  between  $  and  EI  is  the  voltage  induced  by  the  rotor 
flux  leaking  outside  of  the  outer  squirrel  cage.  This  has  the 
m.m.f.  7i  +  72,  and  the  reactance  x\,  thus  is  the  reactance  voltage 
Xi  (I \  +  72).  The  difference  between  E0  and  E  is  the  voltage 
consumed  by  the  primary  impedance :  Z070.  (4)  and  (5)  are  the 
voltages  reduced  to  full  frequency;  the  actual  voltages  are  s 
times  as  high,  but  since  all  three  terms  in  these  equations  are 
induced  voltages,  the  s  cancels. 

21.  From  the  equations  (1)  to  (6)  follows: 

(7) 


(9) 


INDUCTION  MOTOR 


31 


«1  =  1  — 


where : 


,          , 

tt2=    «    (  --  ---  -- 

Vfi       r2       r2 

thus  the  exciting  current: 

/OO    =    YQ£] 

=  E2(g  -  jb)  (01  +  ja2) 
=  ^2(61+^2), 
where  : 


and  the  total  primary  current  is  (3)  :  ' 


where  : 


and  the  primary  impressed  voltage  (6)  : 


where  : 


hence,  absolute: 


(r0 


=  ai  +  r0cj  - 

+  or0C] 


io  =  e2\/ci2  +  c22. 


(10) 


(11) 
(12) 

(13) 
(14) 


(15) 
(16) 

(17) 
(18) 


22.  The  torque  of  the  two  squirrel  cages  is  given  by  the  product 
of  current  and  induced  voltage  in  phase  with  it,  as: 


fj 


hi' 


(19) 


(20) 


32  ELECTRICAL  APPARATUS 

hence,  the  total  torque  : 

D  =  D2  +  Di,  (21), 

and  the  power  output: 

p  =  (i  _  s)  D.  (22) 

(Herefrom  subtracts  the  friction  loss,  to  give  the  net  power 
output.) 

The  power  input  is  : 

Po-=/#o,  V 

(23) 


and  the  volt-ampere  input  : 

Q     = 

E> 

Herefrom  then  follows  the  power-factor  -£t  the  torque  em- 

D  D        - 

ciency  p-,  the  apparent  torque  efficiency  -pr,  the  power  efficiency 

*4)  v*J 

P  p 

•p-  and  the  apparent  power  efficiency  ^- 

23.  As  illustrations  are  shown,  in  Figs.  14  and  15,  the  speed 
curves  and  the  load  curves  of  a  double  squirrel-cage  induction 
motor,  of  the  constants  : 

eQ  =  110  volts; 
Zo  =  0.1  +  0.3J; 
Z,  =  0.5  +  0.2  j; 
Z2  =  0.08  +  0.4  j; 
Fo  =  0.01  -  0.1  j] 

the  speed  curves  for  the  range  from  s  =  0  to  s  =  2,  that  is,  from 
synchronism  to  backward  rotation  at  synchronous  speed.  The 
total  torque  as  well  as  the  two  individual  torques  are  shown  on 
the  speed  curve.  These  curves  are  derived  by  calculating,  for 
the  values  of  s: 

s  =  0,  0.01,  0.02,  0.05,  0.1,  0.15,  0.2,  0.3, 

0.4,  0.6,  0.8,  1.0,  1.2,  1.4,  1.6,  1.8,  2.0, 


INDUCTION  MOTOR 


33 


DOUBLE  SQUIRREL  CAGE 

INDUCTION   MOTOR 

SPEED  CURVES 


-1.0-.9-8  -.7 -.6  -.5-4  -.3  -.2  -.1      0     .1     .2     .3     .4     .5     .6     .7    .8     .9     1.0 
FIG.  14. — Speed  curves  of  double  squirrel-cage  induction  motor. 


I 


DOUBLE  SQUIRREL  CAGE 

INDUCTION  MOTOR 

LOAD  CURVES 


15  20  25  30  35  40  45  50  55  60  KW 


FIG.  15. — Load  curves  of  double  squirrel-cage  induction  motor. 


34  ELECTRICAL  APPARATUS 

the  values: 

_ 
Ct  i    —    J_ 


dig  -f  a25, 
«20  -  aib, 
s        *    . 


r0ci  —  x0c2, 
r0c2 


-f  c22, 


Z)    =    Z>!    +    />2, 

P  =  (i  -  «)  A 

Po  =  e 


and: 

P^     L>    P    ^   Po 

PO'PO'Q'Q'  «' 

Triple  Squirrel-cage  Induction  Motor 
24.  Let: 

$  =  flux,  E  —  voltage,  /  =  current,  and  Z  =  r  +  jx  =  self- 
inductive  impedance,  at  full  frequency  and  reduced  to  primary 
circuit,  and  let  the  quantities  of  the  innermost  squirrel  cage  be 
denoted  by  index  3,  those  of  the  middle  squirrel  cage  by  2,  of 
the  outer  squirrel  cage  by  1,  of  the  primary  circuit  by  0,  and  the 
mutual  inductive  quantities  without  index. 

Also  let  :  F0  =  g  —  jb  =  primary  exciting  admittance. 

It  is  then,  at  slip  s: 
current  in  the  innermost  squirrel  cage: 

(i) 


INDUCTION  MOTOR  35 

current  in  the  middle  squirrel  cage: 

h  =  ~;  (2) 

current  in  the  outer  squirrel  cage: 

f'i 
primary  current: 

/O    =    /3   +   h   +   /I   +    Fo#.  (4) 

The  voltages  are  related  by: 

T7T         t    _       77T        |         F7      T  /Q\ 

where  23  is  the  reactance  due  to  the  flux  leakage  between  the 
third  and  the  second  squirrel  cage;  x2  the  reactance  of  the  leak- 
age flux  between  second  and  first  squirrel  cage;  x\  the  reactance 
of  the  first  squirrel  cage  and  x0  that  of  the  primary  circuit,  that 
is,  #1  +  x0  corresponds  to  the  total  leakage  flux  between  primary 
and  outer  most  squirrel  cage. 

$3,  EZ  and  EI  are  the  true  induced  voltages  in  the  three  squirrel 
cages,  E  the  mutual  inductive  voltage  between  primary  and 
secondary,  and  E0  the  primary  impressed  voltage. 

25.  From  equations  (1)  to  (8)  then  follows: 

(9) 


=  Es(a1-\-ja2))  (11) 

where : 


1  - 


=  s  (—  +  —  +  ^ 
Vr2       r3       r3 


(12) 


36 


ELECTRICAL  APPARATUS 


r2r3 


r2         r3 


1 
/  J 


where : 


&1 

b2 


SXj          SXi 

-  +  -- 


thus  the  exciting  current: 

/oo  =  Y 
=  ^3 

=   ^3 

where  : 


-f  b2b, 


and  the  total  primary  current,  by  (4)  : 


/  2 


where : 


Zo/ 


+  jd2)  (r0  +  j 


where : 


thus,  the  primary  impressed  voltage,  by  (8) : 

E0  =  E,  (6,  +  jb2  +  /,  +  j/2) 


=    E3  ((/,    +  jflTj), 


where: 


(14) 


(15) 


(16) 
(17) 


—J  -T—-TC1  -f-JC2 


(18) 
(19) 

(20) 
(21) 

(22) 
(23) 


INDUCTION  MOTOR  37 

hence,  absolute: 

=>  (24) 

_02!_ 
+  d22,  (25) 

«2#32  /o^ 

— f-> 

e>  =  esV^To?-  (27) 

26.  The  torque  of  the  innermost  squirrel  cage  thus  is : 

Dt  =  S^;  (28) 

that  of  the  middle  squirrel  cage : 

v,  =  ^;  (29) 

7*2 

and  that  of  the  outer  squirrel  cage : 

fli  = s^;  (30) 

the  total  torque  of  the  triple  squirrel-cage  motor  thus  is : 

D  =  D1  +  D2  +  Z>8,  (31) 

and  the  power: 

P  =  (1  -  «)  A  (32) 

the  power  input  is : 

PQ  =  /EQ,  /o/' 

=  e32  (d,flfi  +  d202),  *  (33) 

and  the  volt-ampere  input : 

Q  =  eoi'o.  (34) 

:p 

Herefrom  then  follows  the  power-factor  -^»  the  torque  effi- 
ciency  p-,  apparent  torque  efficiency  T>>  power  efficiency  -5- 

JL  o  v^  *  0 

p 

and  apparent  power  efficiency  -*• 

27.  As  illustrations  are  shown,  in  Figs.  16  and  17,  the  speed 
and  the  load  curves  of  a  triple  squirrel-cage  motor  with  the 
constants : 

€o  =  110  volts; 
ZQ  =  0.1  +0.3J; 
Zl  =  0.8  +0.1J; 
Z2  =  0.2  +0.3J; 
Z3  =  0.05  +  0.8  j; 
Fo  =  0.01  -  0.1  j; 


38 


ELECTRICAL  APPARATUS 


i! 


TRIPLE  SQUIRREL  CAGE 

INDUCTION  MOTOR 

SPEED  CURVES 


-1.0-9  -.8  -.7  -.6  -.5  -.4  -.3  -.2  -.1      0     .1    .2    .3     .4     .5    .6     .7     .8 

FIG.  16. — Speed  curves  of  triple  squirrel-cage  induction  motor. 


FRIPLE  SQUIRREL  OAGE 

INDUCTION  MOTOR 

LOAD  CURVES 


FIG.  17. — Load  curves  of  triple  squirrel-cage  induction  motor. 


INDUCTION  MOTOR 


39 


the  speed  curves  are  shown  from  «  =  0  to  s  =  2,  and  on  them, 
the  individual  torques  of  the  three  squirrel  cages  are  shown  in 
addition  to  the  total  torque. 

These  numerical  values  are  derived  by  calculating,  for  the 

values  of  s: 

8  =  0,  0.01,  0.02,  0.05,  0.1,  0.15,  0.20,  0.30, 
0.40,  0.60,  0.80,  1.0,  1.2,  1.4,  1.6,  1.8,  20, 
the  values: 

„   2    -  .     f°2 


(*+•*.+*) 

Vr2       r3       IV 


&1 


b»  = 


C2   = 


<r2 
ai  — 


SXidi 


r2r3 

SXi     ,     SX} 


$  -f  bib, 

11  +  -  +  ~  +  *i, 
i        r2       r3 


+ 


«  xs    i  •£.    • 

r2r3       r3 


/i  = 

/2    = 

01    = 


63, 

io 


+ 


=  e32  (oi! 


r3 
«22), 


ri 


D  =  D!  +  D2  +  £>: 
p  =  (1  -  «)  D, 

Po  = 

<r- 

and 

P     D    P    D    Po 


CHAPTER  III 

CONCATENATION 

Cascade  or  Tandem  Control  of  Induction  Motors 

28.  If  of  two  induction  motors  the  secondary  of  the  first  motor 
is  connected  to  the  primary  of  the  second  motor,  the  second 
machine  operates  as  a  motor  with  the  voltage  and  frequency 
impressed  upon  it  by  the  secondary  of  the  first  machine.  The 
first  machine  acts  as  general  alternating-current  transformer 
or  frequency  converter  (see  Chapter  XII),  changing  a  part  of  the 
primary  impressed  power  into  secondary  electrical  power  for 
the  supply  of  the  second  machine,  and  a  part  into  mechanical 
work. 

The  frequency  of  the  secondary  voltage  of  the  first  motor,  and 
thus  the  frequency  impressed  upon  the  second  motor,  is  the  fre- 
quency of  slip  below  synchronism,  s.  The  frequency  of  the 
secondary  of  the  second  motor  is  the  difference  between  its  im- 
pressed frequency,  s,  and  its  speed.  Thus,  if  both  motors  are 
connected  together  mechanically,  to  turn  at  the  same  speed, 
1  --  s,  and  have  the  same  number  of  poles,  the  secondary  fre- 
quency of  the  second  motor  is  2  s  -  -  1,  hence  equal  to  zero  at 
s  =  0.5.  That  is,  the  second  motor  reaches  its  synchronism  at 
half  speed.  At  this  speed,  its  torque  becomes  zero,  the  power 
component  of  its  primary  current,  and  thus  the  power  com- 
ponent of  the  secondary  current  of  the  first  motor,  and  thus  also 
the  torque  of  the  first  motor  becomes  zero.  That  is,  a  system  of 
two  concatenated  equal  motors,  with  short-circuited  secondary 
of  the  second  motor,  approaches  half  synchronism  at  no-load, 
in  the  same  manner  as  a  single  induction  motor  approaches 
synchronism.  With  increasing  load,  the  slip  below  half  syn- 
chronism increases. 

In  reality,  at  half  synchronism,  s  =  0.5,  there  is  a  slight  torque 
produced  by  the  first  motor,  as  the  hysteresis  energy  current  of 
the  second  motor  comes  from  the  secondary  of  the  first  motor, 
and  therein,  as  energy  current,  produces  a  small  torque. 

More  generally,  any  pair  of  induction  motors  connected  in 
concatenation  divides  the  speed  so  that  the  sum  of  their  two 

40 


CONCATENATION  41 

respective  speeds  approaches  synchronism  at  no-load;  or,  still 
more  generally,  any  number  of  concatenated  induction  motors 
run  at  such  speeds  that  the  sum  of  their  speeds  approaches 
synchronism  at  no-load. 

With  mechanical  connection  between  the  two  motors,  con- 
catenation thus  offers  a  means  of  operating  two  equal  motors  at 
full  efficiency  at  half  speed  in  tandem,  as  well  as  at  full  speed, 
in  parallel,  and  thereby  gives  the  same  advantage  as  does  series 
parallel  control  with  direct-current  motors. 

With  two  motors  of  different  number  of  poles,  rigidly  con- 
nected together,  concatenation  allows  three  speeds:  that  of  the 
one  motor  alone,  that  of  the  other  motor  alone,  and  the  speed  of 
concatenation  of  both  motors.  Such  concatenation  of  two  motors 
of  different  numbers  of  poles,  has  the  disadvantage  that  at  the 
two  highest  speeds  only  one  motor  is  used,  the  other  idle,  and  the 
apparatus  economy  thus  inferior.  However,  with  certain  ratios 
of  the  number  of  poles,  it  is  possible  to  wind  one  and  the  same 
motor  structure  so  as  to  give  at  the  same  time  two  different 
numbers  of  poles:  For  instance,  a  four-polar  and  an  eight- 
polar  winding;  and  in  this  case,  one  and  the  same  motor  struc- 
ture can  be  used  either  as  four-polar  motor,  with  the  one  winding, 
or  as  eight-polar  motor,  with  the  other  winding,  or  in  concatena- 
tion of  the  two  windings,  corresponding  to  a  twelve-polar  speed. 
Such  " internally  concatenated"  motors  thus  give  three  different 
speeds  at  full  apparatus  economy.  The  only  limitation  is,  that 
only  certain  speeds  and  speed  ratios  can  economically  be  produced 
by  internal  concatenation. 

29.  At  half  synchronism,  the  torque  of  the  concatenated  couple 
of  two  equal  motors  becomes  zero.  Above  half  synchronism, 
the  second  motor  runs  beyond  its  impressed  frequency,  that  is, 
becomes  a  generator.  In  this  case,  due  to  the  reversal  of  current 
in  the  secondary  of  the  first  motor  (this  current  now  being  out- 
flowing or  generator  current  with  regards  to  the  second  motor) 
its  torque  becomes  negative  also,  that  is,  the  concatenated  couple 
becomes  an  induction  generator  above  half  synchronism.  When 
approaching  full  synchronism,  the  generator  torque  of  the  second 
motor,  at  least  if  its  armature  is  of  low  resistance,  becomes  very 
small,  as  this  machine  is  operating  very  far  above  its  synchronous 
speed.  With  regards  to  the  first  motor,  it  thus  begins  to  act 
merely  as  an  impedance  in  the  secondary  circuit,  that  is,  the  first 
machine  becomes  a  motor  again.  Thus,  somewhere  between 


42  ELECTRICAL  APPARATUS 

half  synchronism  and  synchronism,  the  torque  of  the  first  motor 
becomes  zero,  while  the  second  motor  still  has  a  small  negative  or 
generator  torque.  A  little  above  this  speed,  the  torque  of  the 
concatenated  couple  becomes  zero — about  at  two-thirds  syn- 
chronism with  a  couple  of  low-resistance  motors — and  above 
this,  the  concatenated  couple  again  gives  a  positive  or  motor 
torque — though  the  second  motor  still  returns  a  small  negative 
torque — and  again  approaches  zero  at  full  synchronism.  Above 
full  synchronism,  the  concatenated  couple  once  more  becomes 
generator,  but  practically  only  the  first  motor  contributes  to  the 
generator  torque  above  and  the  motor  torque  below  full  syn- 
chronism. Thus,  while  a  concatenated  couple  of  induction 
motors  has  two  operative  motor  speeds,  half  synchronism  and 
full  synchronism,  the  latter  is  uneconomical,  as  the  second  motor 
holds  back,  and  in  the  second  or  full  synchronism  speed  range,  it 
is  more  economical  to  cut  out  the  second  motor  altogether,  by 
short-circuiting  the  secondary  terminals  of  the  first  motor. 

With  resistance  in  the  secondary  of  the  second  motor,  the 
maximum  torque  point  of  the  second  motor  above  half  syn- 
chronism is  shifted  to  higher  speeds,  nearer  to  full  synchronism, 
and  thus  the  speed  between  half  and  full  synchronism,  at  which 
the  concatenated  couple  loses  its  generator  torque  and  again 
becomes  motor,  is  shifted  closer  to  full  synchronism,  and  the 
motor  torque  in  the  second  speed  range,  below  full  synchronism, 
is  greatly  reduced  or  even  disappears.  That  is,  with  high  resist- 
ance in  the  secondary  of  the  second  motor,  the  concatenated 
couple  becomes  generator  or  brake  at  half  synchronism,  and 
remains  so  at  all  higher  speeds,  merely  loses  its  braking  torque 
when  approaching  full  synchronism,  and  regaining  it  again  beyond 
full  synchronism. 

The  speed  torque  curves  of  the  concatenated  couple,  shown  in 
Fig.  18,  with  low-resistance  armature,  and  in  Fig.  19,  with  high 
resistance  in  the  armature  or  secondary  of  the  second  motor, 
illustrate  this. 

30.  The  numerical  calculation  of  a  couple  of  concatenated 
induction  motors  (rigidly  connected  together  on  the  same  shaft 
or  the  equivalent)  can  be  carried  out  as  follows: 

Let: 

n  =  number  of  pairs  of  poles  of  the  first  motor, 
n'  =  number  of  pairs  of  poles  of  the  second  motor, 


CONCATENATION  43 

a  =  —  =  ratio  of  poles,  (1) 

'       /  =  supply  frequency. 

Full  synchronous  speed  of  the  first  motor  then  is: 

«„  =  {;  (2) 

of  the  second  motor: 

Cf  J  .  fQ\ 

f? 

At  slip  s  and  thus  speed  ratio  (1  —  s)  of  the  first  motor,  its 
speed  is: 

S  =  (1  -  s)  SQ  =  (1  -  s)  £,  (4) 

and  the  frequency  of  its  secondary  circuit,  and  thus  the  frequency 
of  the  primary  circuit  of  the  second  motor: 

synchronous  speed  of  the  second  motor  at  this  frequency  is : 

•*-•*?.  .  V: 

the  speed  of  the  second  motor,  however,  is  the  same  as  that  of 
the  first  motor,  S, 

hence,  the  slip  of  speed  of  the  second  motor  below  its  synchronous 
speed,  is: 


n  n 

and  the  slip  of  frequency  thus  is: 


s'  =  s  (1  +  a)  -  a.  (5) 

This  slip  of  the  second  motor,  s',  becomes  zero,  that  is,  the 
couple  reaches  the  synchronism  of  concatenation,  for: 


44  ELECTRICAL  APPARATUS 

The  speed  in  this  case  is : 


So"  =  (1  -  so)  (7) 

IV 

f 


31.  If: 

a  =  1, 

that  is,  two  equal  motors,  as  for  instance  two  four-polar  motors 

n  =  nf  =  4, 
it  is: 

so  =  0.5, :': 

' 


while  at  full  synchronism: 

.  /_/. 


n       ¥ 


If: 


n  =  4, 
n'-8, 

it  is: 

2 


that  is,  corresponding  to  a  twelve-polar  motor. 
While: 

a  /          / 

^0  =  n  =  4' 
if: 

a  =  0.5, 
n  =  8, 
nr  =  4, 
it  is: 

1 


/  / 

-" 


CONCATENATION  45 

that  is,  corresponding  to  a  twelve-polar  motor  again.  That  is, 
as  regards  to  the  speed  of  the  concatenated  couple,  it  is  immaterial 
in  which  order  the  two  motors  are  concatenated. 

32.  It  is  then,  in  a  concatenated  motor  couple  of  pole  ratio : 


a  —  — 
n 


if: 


s  =  slip  of  first  motor  below  full  synchronism. 

The  primary  circuit  of  the  first  motor  is  of  full  frequency. 

The  secondary  circuit  of  the  first  motor  is  of  frequency  s. 

The  primary  circuit  of  the  second  motor  is  of  frequency  s. 

The  secondary  circuit  of  the  second  motor  is  of  frequency  s'  = 
s  (1  +  a)  -  a. 

Synchronism  of  concatenation  is  reached  at  : 


Let  thus: 

CQ  =  voltage  impressed  of  first  motor  primary; 
YQ  =  g  —  jb       =  exciting  admittance  of  first  motor; 
Y'o  =  gf  —  jbf     —  exciting  admittance  of  second  motor; 
Zo  =  rQ   +  jxo  =  self-inductive    impedance     of     first     motor 

primary; 
Z'o  =  r'0  +  JX'Q  =  self-inductive   impedance    of    second  motor 

primary; 
Zi  =  ri  +  jxi  =  self  -inductive  impedance  of  first  motor  second- 

ary; 
Z'i  =  r'i  +  jx\  =  self-inductive   impedance    of   second   motor 

secondary. 

Assuming  all  these  quantities  reduced  to  the  same  number  of 
turns  per  circuit,  and  to  full  frequency,  as  usual. 

If: 

e    =  counter  e.m.f  .  generated  in  the  second  motor  by  its  mutual 
magnetic  flux,  reduced  to  full  frequency. 

It  is  then  : 
secondary  current  of  second  motor: 

,,  s'e  [s  (1  -f  a)  —  a]  e  , 

1  '  =  PTTjWi  =  r>,  +  j  [.  (1  +  a)  -  a]  X'\  =  e^ 


46  ELECTRICAL  APPARATUS 

where: 


[s  (1  +  o)  -  a]      ) 

(9) 


m 

x'i  [s  (1  +  a)  -  a]2 


m  =  r'i2  +  zV  (s  (1  +  a)  -  a)2;  (10) 

exciting  current  of  second  motor: 

/'„„  =  eY'  =  e  (gr  -  jb'),  (11) 

hence,  primary  current  of  second  motor,  and  also  secondary 
current  of  first  motor: 

/'o  =  /i  =  /']  +  /'oo 

=  e  (b,  -  jb,),  (12) 

where  : 

bi**&i  +  g',  (  ^ 

b2  =  a2  +  V, 

the  impedance  of  the  circuit  comprising  the  primary  of  the 
second,  and  the  secondary  of  the  first  motor,  is: 

Z  =  Zs  +  ZV  =  (n  +  /o)  +  js  (x,  +  »'0),-         (14) 

hence,  the  counter  e.m.f.,  or  induced  voltage  in  the  secondary 
of  the  first  motor,  of  frequency  is  : 

8$  i  =  se  +  /iZ, 
hence,  reduced  to  full  frequency: 


where : 

ci  =  1  + 


33.  The  primary  exciting  current  of  the  first  motor  is : 

/oo  =  E,Y 

=  e(dl-  jd*),  (17) 

where : 


=  e 
r    -4-  r' 

(Ci  +  JC2), 

+  (xi  +  a/o)  62 

(15) 
(16) 

s 

62 

CONCATENATION  47 

thus,  the  total  primary  current  of  the  first  motor,  or  supply 
current : 

/o  =  /i  ~h  /oo 

=  e-(fi-jf*),  (19) 

where : 

/,  =  61  + 

/2    =    &2   + 

and  the  primary  impressed  voltage  of  the  first  motor,  or  supply 
voltage : 

EQ    =    EI   +   Zo/0 

-  e  foi  +  J«r.),  (21) 

where : 

<7i  =  Ci  +  ro/i  +  -To/2  /22) 

and,  absolute: 


_  __ 

eo  =  e      ^M-T2"2>  (23) 

thus  : 

e  =  --^=-  (24) 


Substituting  now  this  value  of  e  in  the  preceding,  gives  the 
values  of  the  currents  and  voltages  in  the  different  circuits. 
34.  It  thus  is,  supply  current  : 


power  input: 

P.  = 


,  ,/lgi   - 


volt-ampere  input: 


and  herefrom  power-factor,  etc. 
The  torque  of  the  second  motor  is  : 


T'  = 

= 

The  torque  of  the  first  motor  is  : 

n  =  /#i,  /«/' 

=  e2  (ci/i  -  Cj/2), 


48  ELECTRICAL  APPARATUS 

hence,  the  total  torque  of  the  concatenated  couple : 

T  =  T'  +  T!  =  e2  (a}  +  c,/i  -  c2/2), 
and  herefrom  the  power  output : 

P  =  (1  -  s)  T, 

thus  the  torque  and  power  efficiencies  and  apparent  efficiencies, 
etc. 

35.  As  instances  are.  calculated,  and  shown  in  Fig.  18,  the  speed 


+  1.4    1.3    1.2     1.1     1.0    0.9    0.8     0.7    0.6    0.5    0.4    0.3    0.2    0.1      0-0.1-0.2-0.3-0.4-0.5-0.6-0.7 

FIG.  18. — Speed  torque  curves  of  concatenated  couple  with  low  resistance 

secondary. 

torque  curves  of  the  concatenated  couple  of  two  equal  motors: 
a  =  1,  of  the  constants :  eo  =  110  volts. 

Y  =  Y'  =  0.01  -  0.1  j; 
Z0  =  Z'0  =  0.1  +  0.3  j- 
Zl  =  Z'l  =  0.1  +  0.3  j. 

Fig.  18  also  shows,  separately,  the  torque  of  the  second  motor, 
and  the  supply  current. 

Fig.  19  shows  the  speed  torque  curves  of  the  same  concate- 
nated couple  with  an  additional  resistance  r  =  0.5  inserted  into 
the  secondary  of  the  second  motor. 

The  load  curves  of  the  same  motor,  Fig.  18,  for  concatenated 
running,  and  also  separately  the  load  curves  of  either  motor, 


CONCATENATION 


49 


are  given  on  page  358  of  "  Theoretical  Elements  of  Electrical 
Engineering." 

36.  It  is  possible  in  concatenation  of  two  motors  of  different 
number  of  poles,  to  use  one  and  the  same  magnetic  structure  for 
both  motors.  Suppose  the  stator  is  wound  with  an  n-polar 
primary,  receiving  the  supply  voltage,  and  at  the  same  time  with 
an  n(  polar  short-circuited  secondary  winding.  The  rotor  is 
wound  with  an  n-polar  winding  as  secondary  to  the  n-polar 
primary  winding,  but  this  n-polar  secondary  winding  is  not 
short-circuited,  but  connected  to  the  terminals  of  a  second 


-cooo 

-4000 
-2000 
0 
-2000 
-4000 

-cooe 

*  —  ^ 

d 

N 

ZG-Z 

.0.1- 

-0.3J 

Y  - 

).01  - 

0.1J 
2 

fe 

*^>s 

\ 

1 

2 

z 

£ 

^ 

^r=^. 

\\ 

S? 

V 

~~/ 

/ 

1.0     ( 

.9       0 

8       0 

7       0 

G       0 

x, 
.5       0 

S 
4       0 

3      0 

2        0 

1      0.< 

FIG.  19. — Speed-torque  curves  of  concatenated  couple  with  resistance  in 

second  secondary. 

n'-polar  winding,  also  located  on  the  rotor.  This  latter  thus 
receives  the  secondary  current  from  the  n-polar  winding  and 
acts  as  n'-polar  primary  to  the  short-circuited  stator  winding  as 
secondary.  This  gives  an  n-polar  motor  concatenated  to  an 
n'-polar,  and  the  magnetic  structure  simultaneously  carries  an 
n-polar  and  an  n'-polar  magnetic  field.  With  this  arrangement 
of  "internal  concatenation,"  it  is  essential  to  choose  the  number 
of  poles,  n  and  n',  so  that  the  two  rotating  fields  do  not  interfere 
with  each  other,  that  is,  the  n'-polar  field  does  not  induce  in  the 
n-polar  winding,  nor  the  n-polar  field  in  the  n'-polar  winding. 
This  is  the  case  if  the  one  field  has  twice  as  many  poles  as  the 
other,  for  instance  a  four-polar  and  an  eight-polar  field. 

If  such  a  fractional-pitch  winding  is  used,  that  the  coil  pitch 
is  suited  for  an  n-polar  as  well  as  an  n'-polar  winding,  then  the 
same  winding  can  be  used  for  both  sets  of  poles.  In  the  stator, 
the  equipotential  points  of  a  2p-polar  winding  are  points  of 
opposite  polarity  of  a  p-polar  winding,  and  thus,  by  connecting 
together  the  equipotential  points  of  a  2  p-polar  primary  winding, 


50  ELECTRICAL  APPARATUS 

this  winding  becomes  at  the  same  time  a  p-polar  short-circuited 
winding.  On  the  rotor,  in  some  slots,  the  secondary  current  of 
the  n-polar  and  the  primary  current  of  the  n'-polar  winding  flow 
in  the  same  direction,  in  other  slots  flow  in  opposite  direction, 
thus  neutralize  in  the  latter,  and  the  turns  can  be  omitted  in 
concatenation — but  would  be  put  in  for  use  of  the  structure  as 
single  motor  of  n,  or  of  n'  poles,  where  such  is  desired.  Thus, 
on  the  rotor  one  single  winding  also  is  sufficient,  and  this  arrange- 
ment of  internal  concatenation  with  single  stator  and  single  rotor 
winding  thus  is  more  efficient  than  the  use  of  two  separate  motors, 
and  gives  somewhat  better  constants,  as  the  self-inductive  im- 
pedance of  the  rotor  is  less,  due  to  the  omission  of  one-third  of 
the  turns  in  which  the  currents  neutralize  (Hunt  motor). 

The  disadvantage  of  this  arrangement  of  internal  concatenation 
with  single  stator  and  rotor  winding  is  the  limitation  of  the  avail- 
able speeds,  as  it  is  adapted  only  to  4  -5-  8  -*-  12  poles  and 
multiples  thereof,  thus  to  speed  ratios  of  1  -r-  J^  -r-  J^,  the  last 
being  the  concatenated  speed. 

Such  internally  concatenated  motors  may  be  used  advantage- 
ously sometime  as  constant-speed  motors,  that  is,  always  run- 
ning in  concatenation,  for  very  slow-speed  motors  of  very  large 
number  of  poles. 

37.  Theoretically,  any  number  of  motors  may  be  concatenated. 
It  is  rarely  economical,  however,  to  go  beyond  two  motors  in 
concatenation,  as  with  the  increasing  number  of  motors,  the 
constants  of  the  concatenated  system  rapidly  become  poorer. 

If: 

Fo  -  g  -  jb, 
ZQ  =  r0  H-  jx0) 
Zi  =  ri+  jxi, 

are  the  constants  of  a  motor,  and  we  denote : 

Z  =  ZQ  +  Zl  =  (r0  +  ri)  +  j  (xo  +  Si) 
=  r  +  jx 

then  the  characteristic  constant  of  this  motor — which  char- 
acterizes its  performance — is: 

0  =  yz\ 

if  now  two  such  motors  are  concatenated,  the  exciting  admittance 
of  the  concatenated  couple  is  (approximately) : 

r  =  2  F, 


CONCATENATION  51 

as  the  first  motor  carries  the  exciting  current  of  the  second 
motor. 

The  total  self-inductive  impedance  of  the  couple  is  that  of 
both  motors  in  series : 

Z'  =  2Z; 

thus  the  characteristic  constant  of  the  concatenated  couple  is: 

0'  =  y'z' 
=  4yz 
=  40, 

that  is,  four  times  as  high  as  in  a  single  motor;  in  other  words, 
the  performance  characteristics,  as  power-factor,  etc.,  are  very 
much  inferior  to  those  of  a  single  motor. 

With  three  motors  in  concatenation,  the  constants  of  the 
system  of  three  motors  are : 

F"  =  3  Y, 
Z"  =  3  Z, 

thus  the  characteristic  constant: 

&"  =  y"z" 
=  9yz 
=  90, 

or  nine  times  higher  than  in  a  single  motor.  In  other  words, 
the  characteristic  constant  increases  with  the  square  of  the 
number  of  motors  in  concatenation,  and  thus  concatenation 
of  more  than  two  motors  would  be  permissible  only  with  motors 
of  very  good  constants. 

The  calculation  of  a  concatenated  system  of  three  or  more 
motors  is  carried  out  in  the  same  manner  as  that  of  two  motors, 
by  starting  with  the  secondary  circuit  of  the  last  motor,  and 
building  up  toward  the  primary  circuit  of  the  first  motor. 


CHAPTER  IV 
INDUCTION   MOTOR   WITH   SECONDARY   EXCITATION 

38.  While  in  the  typical  synchronous  machine  and  commu- 
tating  machine  the  magnetic  field  is  excited  by  a  direct  current, 
characteristic  of  the  induction  machine  is,  that  the  magnetic 
field  is  excited  by  an  alternating  current  derived  from  the  alter- 
nating supply  voltage,  just  as  in  the  alternating-current  trans- 
former. As  the  alternating  magnetizing  current  is  a  wattless 
reactive  current,  the  result  is,  that  the  alternating-current  input 
into  the  induction  motor  is  always  lagging,  the  more  so,  the 
larger  a  part  of  the  total  current  is  given  by  the  magnetizing 
current.  To  secure  good  power-factor  in  an  induction  motor, 
the  magnetizing  current,  that  is,  the  current  which  produces 
the  magnetic  field  flux,  must  be  kept  as  small  as  possible.  This 
means  as  small  an  air  gap  between  stator  and  rotor  as  mechanic- 
ally permissible,  and  as  large  a  number  of  primary  turns  per  pole, 
that  is,  as  large  a  pole  pitch,  as  economically  permissible. 

In  motors,  in  which  the  speed — compared  to  the  motor  out- 
put— is  not  too  low,  good  constants  can  be  secured.  This, 
however,  is  not  possible  in  motors,  in  which  the  speed  is  very 
low,  that  is,  the  number  of  poles  large  compared  with  the  out- 
put, and  the  pole  pitch  thus  must  for  economical  reasons  be  kept 
small — as  for  instance  a  100-hp.  60-cycle  motor  for  90  revolu- 
tions, that  is,  80  poles — or  where  the  requirement  of  an  excessive 
momentary  overload  capacity  has  to  be  met,  etc.  In  such  motors 
of  necessity  the  exciting  current  or  current  at  no-load — which 
is  practically  all  magnetizing  current — is  a  very  large  part  of 
full-load  current,  and  while  fair  efficiencies  may  nevertheless  be 
secured,  power-factor  and  apparent  efficiency  necessarily  are 
very  low. 

As  illustration  is  shown  in  Fig.  20  the  load  curve  of  a  typical 
100-hp.  60-cycle  80-polar  induction  motor  (90  revolutions  per 
minute)  of  the  constants: 

Impressed  voltage:  eQ  =  500. 

Primary  exciting  admittance:  Y0  =  0.02  —  0.6  j. 

Primary  self-inductive  impedance:  ZQ  =  0.1  +  0.3  j. 

Secondary  self -inductive  impedance:  Z\  =  0.1  +  0.3  j. 

52 


INDUCTION  MOTOR 


53 


As  seen,  at  full-load  of  75  kw.  output, 
the  efficiency  is  80  per  cent.,  which  is  fair  for  a  slow-speed  motor. 

But  the  power-factor  is  55  per  cent.,  the  apparent  efficiency 
only  44  per  cent.,  and  the  exciting  current  is  75  per  cent,  of  full- 
load  current. 

This  motor-load  curve  may  be  compared  with  that  of  a  typical 
induction  motor,  of  exciting  admittance: 

Fo  =  0.01  -  0.1  j, 

given  on  page  234  of  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena"  5th  edition,  and  page  319  of  " Theoretical 


LOW  SPEED  INDUCTION   MOTOR 

=  500  Z0=.1+.3j 

Y0=02~. 


IS  10  20  30   40  50   60   70   80  9,0  100  1  0  120  ™,- 

FIG.  20. — Low-speed  induction  motor,  load  curves. 

Elements  of  Electrical  Engineering,"  4th  edition,  to  see  the 
difference. 

39.  In  the  synchronous  machine  usually  the  stator,  in  corn- 
mutating  machines  the  rotor  is  the  armature,  that  is,  the  element 
to  which  electrical  power  is  supplied,  and  in  which  electrical 
power  is  converted  into  the  mechanical  power  output  of  the 
motor.  The  rotor  of  the  typical  synchronous  machine,  and  the 
stator  of  the  commutating  machine  are  the  field,  that  is,  in 
them  no  electric  power  is  consumed  by  conversion  into  mechanical 
work,  but  their  purpose  is  to  produce  the  magnetic  field  flux, 
through  which  the  armature  rotates. 

In  the  induction  machine,  it  is  usually  the  stator,  which  is  the 


54  ELECTRICAL  APPARATUS 

primary,  that  is,  which  receives  electric  power  and  converts  it 
into  mechanical  power,  and  the  primary  or  stator  of  the  induc- 
tion machine  thus  corresponds  to  the  armature  of  the  synchro- 
nous or  commutating  machine.  In  the  secondary  or  rotor  of  the 
induction  machine,  low-frequency  currents — of  the  frequency 
of  slip — are  induced  by  the  primary,  but  the  magnetic  field  flux 
is  produced  by  the  exciting  current  which  traverses  the  primary 
or  armature  or  stator.  Thus  the  induction  machine  may  be 
considered  as  a  machine  in  which  the  magnetic  field  is  produced 
by  the  armature  reaction,  and  corresponds  to  a  synchronous 
machine,  in  which  the  field  coils  are  short-circuited  and  the 
field  produced  by  armature  reaction  by  lagging  currents  in  the 
armature. 

As  the  rotor  or  secondary  of  the  induction  machine  corresponds 
structurally  to  the  field  of  the  synchronous  or  commutating 
machine,  field  excitation  thus  can  be  given  to  the  induction 
machine  by  passing  a  current  through  the  rotor  or  secondary  and 
thereby  more  or  less  relieving  the  primary  of  its  function  of  giv- 
ing the  field  excitation. 

Thus  in  a  slow-speed  induction  motor,  of  very  high  exciting 
current  and  correspondingly  poor  constants,  by  passing  an 
exciting  current  of  suitable  value  through  the  rotor  or  secondary, 
the  primary  can  be  made  non-inductive,  or  even  leading  current 
produced,  or — with  a  lesser  exciting  current  in  the  rotor — at 
least  the  power-factor  increased. 

Various  such  methods  of  secondary  excitation  have  been  pro- 
posed, and  to  some  extent  used. 

1.  Passing  a  direct  current  through  the  rotor  for  excitation. 
In  this  case,  as  the  frequency  of  the  secondary  currents  is  the 

frequency  of  slip,  with  a  direct  current,  the  frequency  is  zero, 
that  is,  the  motor  becomes  a  synchronous  motor. 

2.  Excitation  through  commutator,  by  the  alternating  supply 
current,  either  in  shunt  or  in  series  to  the  armature. 

At  the  supply  frequency,  /,  and  slip,  s,  the  frequency  of  rotation 
and  thus  of  commutation  is  (1  —  s)/,  and  the  full  frequency  cur- 
rents supplied  to  the  commutator  thus  give  in  the  rotor  the 
effective  frequency,/  —  (1  —  s)  /  =  sf,  that  is,  the  frequency  of 
slip,  thus  are  suitable  as  exciting  currents. 

3.  Concatenation  with  a  synchronous  motor. 

If  a  low-frequency  synchronous  machine  is  mounted  on  the 
induction-motor  shaft,  and  its  armature  connected  into  the  indue- 


INDUCTION  MOTOR  55 

tion-motor  secondary,  the  synchronous  machine  feeds  low-fre- 
quency exciting  currents  into  the  induction  machine,  and  thereby 
permits  controlling  it  by  using  suitable  voltage  and  phase. 
If  the  induction  machine  has  n  times  as  many  poles  as  the 

synchronous  machine,  the  frequency  of  rotation  of  the  synchro- 
j  ^  _  g 

nous  machine  is  -  that  of  the  induction  machine,  or  —    —    How- 
n  n 

ever,  the  frequency  generated  by  the  synchronous  machine  must 
be  the  frequency  of  the  induction-machine  secondary  currents, 
that  is,  the  frequency  of  slip  s. 
Hence  : 

1  -  s 


or: 


that  is,  the  concatenated  couple  is  synchronous,  that  is,  runs  at 
constant  speed  at  all  loads,  but  not  at  synchronous  speed,  but  at 

constant  slip  —  ^—  r- 

4.  Concatenation  with  a  low-frequency  commutating  machine. 
If  a  commutating  machine  is  mounted  on  the  induction-motor 

shaft,  and  connected  in  series  into  the  induction-motor  secondary, 
the  commutating  machine  generates  an  alternating  voltage  of  the 
frequency  of  the  currents  which  excite  its  field,  and  if  the  field 
is  excited  in  series  or  shunt  with  the  armature,  in  the  circuit  of 
the  induction  machine  secondary,  it  generates  voltage  at  the 
frequency  of  slip,  whatever  the  latter  may  be.  That  is,  the 
induction  motor  remains  asynchronous,  increases  in  slip  with 
increase  of  load. 

5.  Excitation  by  a  condenser  in  the  secondary  circuit  of  the 
induction  motor. 

As  the  magnetizing  current  required  by  the  induction  motor  is 
a  reactive,  that  is,  wattless  lagging  current,  it  does  not  require  a 
generator  for  its  production,  but  any  apparatus  consuming  lead- 
ing, that  is,  generating  lagging  currents,  such  as  a  condenser,  can 
be  used  to  supply  the  magnetizing  current. 

40.  However,  condenser,  or  synchronous  or  commutating 
machine,  etc.,  in  the  secondary  of  the  induction  motor  do  not 
merely  give  the  magnetizing  current  and  thereby  permit  power- 
factor  control,  but  they  may,  depending  on  their  design  or  appli- 
cation, change  the  characteristics  of  the  induction  machine,  as 
regards  to  speed  and  speed  regulation,  the  capacity,  etc. 


56  ELECTRICAL  APPARATUS 

If  by  synchronous  or  commutating  machine  a  voltage  is 
inserted  into  the  secondary  of  the  induction  machine,  this  vol- 
tage may  be  constant,  or  varied  with  the  speed,  the  load,  the  slip, 
etc.,  and  thereby  give  various  motor  characteristics.  Further- 
more, such  voltage  may  be  inserted  at  any  phase  relation  from 
zero  to  360°.  If  this  voltage  is  inserted  90°  behind  the  secondary 
current,  it  makes  this  current  leading  or  magnetizing  and  so  in- 
creases the  power-factor.  If,  however,  the  voltage  is  inserted 
in  phase  with  the  secondary  induced  voltage  of  the  induction 
machine,  it  has  no  effect  on  the  power-factor,  but  merely  lowers 
the  speed  of  the  motor  if  in  phase,  raises  it  if  in  opposition  to  the 
secondary  induced  voltage  of  the  induction  machine,  and  hereby 
permits  speed  control,  if  derived  from  a  commutating  machine. 
For  instance,  by  a  voltage  in  phase  with  and  proportional  to  the 
secondary  current,  the  drop  of  speed  of  the  motor  can  be  increased 
and  series-motor  characteristics  secured,  in  the  same  manner  as 
by  the  insertion  of  resistance  in  the  induction-motor  secondary. 
The  difference  however  is,  that  resistance  in  the  induction-motor 
secondary  reduces  the  efficiency  in  the  same  proportion  as  it 
lowers  the  speed,  and  thus  is  inefficient  for  speed  control.  The 
insertion  of  an  e.m.f.,  however,  while  lowering  the  speed,  does 
not  lower  the  efficiency,  as  the  power  corresponding  to  the  lowered 
speed  is  taken  up  by  the  inserted  voltage  and  returned  as  output 
of  the  synchronous  or  commutating  machine.  Or,  by  inserting  a 
voltage  proportional  to  the  load  and  in  opposition  to  the  induced 
secondary  voltage,  the  motor  speed  can  be  maintained  constant, 
or  increased  with  the  load,  etc. 

If  then  a  voltage  is  inserted  by  a  commutating  machine  in  the 
induction-motor  secondary,  which  is  displaced  in  phase  by  angle 
a  from  the  secondary  induced  voltage,  a  component  of  this  vol- 
tage: sin  a,  acts  magnetizing  or  demagnetizing,  the  other  com- 
ponent: cos  a,  acts  increasing  or  decreasing  the  speed,  and  thus 
various  effects  can  be  produced. 

As  the  current  consumed  by  a  condenser  is  proportional  to  the 
frequency,  while  that  passing  through  an  inductive  reactance  is 
inverse  proportional  to  the  frequency,  when  using  a  condenser 
in  the  secondary  circuit  of  the  induction  motor,  its  effective  im- 
pedance at  the  varying  frequency  of  slip  is: 


S  =  ri  sx,  - 

where  x2  is  the  capacity  reactance  at  full  frequency. 


INDUCTION  MOTOR  57 

For  s  =  0,  Zis  =  oo  t  that  is,  the  motor  has  no  power  at  or  near 
synchronism. 
For: 

«*  -  *2  -  0, 
s 

or 

S 

it  is: 


and  the  current  taken  by  the  motor  is  a  maximum.  The  power 
output  thus  is  a  maximum  not  when  approaching  synchronism, 
as  in  the  typical  induction  motor,  but  at  a  speed  depending  on  the 
slip, 

§ 

So  =  Vi? 

and  by  varying  the  capacity  reactance,  #2,  various  values  of  reson- 
ance slip,  s0,  thus  can  be  produced,  and  thereby  speed  control  of 
the  motor  secured.  However,  for  most  purposes,  this  is  uneco- 
nomical, due  to  the  very  large  values  of  capacity  required. 

Induction  Motor  Converted  to  Synchronous 

41.  If,  when  an  induction  motor  has  reached  full  speed,  a  direct 
current  is  sent  through  its  secondary  .circuit,  unless  heavily 
loaded  and  of  high  secondary  resistance  and  thus  great  slip,  it 
drops  into  synchronism  and  runs  as  synchronous  motor. 

The  starting  operations  of  such  an  induction  motor  in  conver- 
sion to  synchronous  motor  thus  are  (Fig.  21) : 

First  step:  secondary  closed  through  resistance:  A. 

Second  step:  resistance  partly  cut  out:  B. 

Third  step:  resistance  all  cut  out:  C. 

Fourth  step :  direct  current  passed  through  the  secondary :  D. 

In  this  case,  for  the  last  or  synchronous-motor  step,  usually 
the  direct-current  supply  will  be  connected  between  one  phase 
and  the  other  two  phases,  the  latter  remaining  short-circuited 
to  each  other,  as  shown  in  Fig.  21,  D.  This  arrangement  retains 
a  short-circuit  in  the  rotor — now  the  field — in  quadrature  with 
the  excitation,  which  acts  as  damper  against  hunting  (Danielson 
motor). 


58 


ELECTRICAL  APPARATUS 


In  the  synchronous  motor,  Fig.  21,  D,  produced  from  the  induc- 
tion motor,  Fig.  21,  C,  it  is: 
Let: 

YQ  =  g   —  jb    =  primary  exciting  admittance 

of  the  induction  machine, 
ZQ  =  r0  +  jxo  =  primary  self-inductive  impe- 
dance, 

Zi  =  f]  +  jxi  =  secondary  self-inductive  im- 
pedance. 


FIG.  21. — Starting  of  induction  motor  and   conversion  to  synchronous. 

The  secondary  resistance,  n,  is  that  of  the  field  exciting  winding, 
thus  does  not  further  come  into  consideration  in  calculating  the 
motor  curves,  except  in  the  efficiency,  as  i-?r\  is  the  loss  of  power 
in  the  field,  if  ii  =  field  exciting  current.  x\  is  of  little  further 
importance,  as  the  frequency  is  zero.  It  represents  the  magnetic 
leakage  between  the  synchronous  motor  poles. 

?'o  is  the  armature  resistance  and  XQ  the  armature  self-inductive 
reactance  of  the  synchronous  machine. 

However,  x0  is  net  the  synchronous  impedance,  which  enters 
the  equation  of  the  synchronous  machine,  but  is  only  the  self- 
inductive  part  of  it,  or  the  true  armature  self-inductance.  The 


INDUCTION  MOTOR  59 

mutual  inductive  part  of  the  synchronous  impedance,  or  the 
effective  reactance  of  armature  reaction,  #',  is  not  contained  in  XQ. 

The  effective  reactance  of  armature  reaction  of  the  synchro- 
nous machine,  x',  represents  the  field  excitation  consumed  by  the 
armature  m.m.f.,  and  is  the  voltage  corresponding  to  this  field 
excitation,  divided  by  the  armature  current  which  consumes  this 
field  excitation. 

6,  the  exciting  susceptance,  is  the  magnetizing  armature 
current,  divided  by  the  voltage  induced  by  it,  thus,  x',  the  effect- 
ive reactance  of  synchronous-motor  armature  reaction,  is  the 
reciprocal  of  the  exciting  susceptance  of  the  induction  machine. 

The  total  or  synchronous  reactance  of  the  induction  machine 
as  synchronous  motor  thus  is: 

$   =  XQ  +  X1 

=  x»  +  r 

The  exciting  conductance,  g,  represents  the  loss  by  hysteresis, 
etc.,  in  the  iron  of  the  machine.  As  synchronous  machine,  this 
loss  is  supplied  by  the  mechanical  power,  and  not  electrically, 
and  the  hysteresis  loss  in  the  induction  machine  as  synchronous 
motor  thus  is:  ezg. 

We  thus  have: 

The  induction  motor  of  the  constants,  per  phase : 
Exciting  admittance :  Y o  =  g  —  jb, 

Primary  self-inductive  impedance :         Zo  =  r0  +  JXQ, 
Secondary  self-inductive  impedance:      Zi  =  7*1  +  jx\, 

by  passing  direct  current  through  the  secondary  or  rotor,  be- 
comes a  synchronous  motor  of  the  constants,  per  phase : 

Armature  resistance:  r0, 

Synchronous  impedance:  x  =  XQ  •+•  r!  (1) 

Total  power  consumed  in  field  excitation: 

P  =  2  ;2n,  (2) 

where  i  =  field  exciting  current. 

Power  consumed  by  hysteresis: 

P  =  «V  (3) 


60  ELECTRICAL  APPARATUS 

42.  Let,  in  a  synchronous  motor: 

EQ  =  impressed  voltage, 

E    =  counter    e.m.f.,     or    nominal    induced 

voltage, 

Z    =  r  +  jx  =  synchronous  impedance, 
/    =  i\  —  jit  =  current, 
it  is  then: 

E0  =  E  +  ZI 

=  $  +  (rii  +  xiz)  +  j  (0*1  -  ri2),          (4) 
or: 

E    =  E0-  ZI 

.  =  EQ  -  (rii  +  xiz)  -  j  (xii  -  riz),         (5) 

or,  reduced  to  absolute  values,  and  choosing: 

E    =  e    =  real  axis  in  equation  (4), 
EQ  =  e0  =  real  axis  in  equation  (5), 

e02  =  (e   +  rii  +  xi2)2  +  (atfi  -  n'2)2  [e  =  real  axis],       (6) 
€2    =  (eo  —  n'i  +  xiz)2  +  (xi'i  —  n'2)2  [e0  =  real  axis].       (7) 

Equations  (6)  and  (7)  are  the  two  forms  of  the  fundamental 
equation  of  the  synchronous  motor,  in  the  form  most  convenient 
for  the  calculation  of  load  and  speed  curves. 

In  (7),  i\  is  the  energy  component,  and  iz  the  reactive  com- 
ponent of  the  current  with  respect  to  the  impressed  voltage,  but 
not  with  respect  to  the  induced  voltage;  in  (6),  i\  is  the  energy 
component  and  i2  the  reactive  component  of  the  current  with 
respect  to  the  induced  voltage,  but  not  with  respect  to  the 
impressed  voltage. 

The  condition  of  motor  operation  at  unity  power-factor  is : 

iz  =  0  in  equation  (7). 
Thus: 

e2  =  (60  -  ri)2  +  aW  (8) 

at  no-load,  for  i\  =  0,  this  gives:  e  =  eQ,  as  was  to  be  expected- 
Equation  (8)  gives  the  variation  of  the  induced  voltage  and 
thus  of  the  field  excitation,  required  to  maintain  unity  power- 
factor  at  all  loads,  that  is,  currents,  i\. 
From  (8)  follows: 


INDUCTION  MOTOR  61 

Thus,  the  minimum  possible  value  of  the  counter  e.m.f.,  e, 
is  given  by  equating  the  square  root  to  zero,  as : 


x 

e  =  - 


For  a  given  value  of  the  counter  e.m.f.,  e,  that  is,  constant 
field  excitation,  it  is,  from  (7) : 


xe0  .  .    . 

*2  =  -  ±     "    "     ' 


or,  if  the  synchronous  impedance,  x}  is  very  large  compared  with 
r,  and  thus,  approximately  : 


The  maximum  value,  which  the  energy  current,  ^i,  can  have, 
at  a  given  counter  e.m.f.,  e,  is  given  by  equating  the  square  root 
to  zero,  as: 


For:  ij  =  0,  or  at  no-load,  it  is,  by  (11): 

.    _  ep  ±  e 
~V 

Equations  (9)  and  (12)  give  two  values  of  the  currents  ii 
and  iz,  of  which  one  is  very  large,  corresponds  to  the  upper  or 
unstable  part  of  the  synchronous  motor-power  characteristics 
shown  on  page  325  of  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena,"  5th  edition. 

43.  Denoting,  in  equation  (5)  : 

E  =  e'  -  je",  (13) 

and  again  choosing  EQ  =  eQ,  as  the  real  axis,  (5)  becomes  : 

e'  —  j^'  —  («o  —  rii  —  xiz)  —  j  (xii  —  n*2),  (14) 

and  the  electric  power  input  into  the  motor  then  is  : 

Po  =  /E0,  II' 

=  eQilt  (15) 

the  power  output  at  the  armature  conductor  is  : 


62  ELECTRICAL  APPARATUS 

hence  by  (14): 

Pi  =  ii  Oo  -  rii  -  xiz)  +  iz  (xi\  -  n'2),  (16) 

expanded,  this  gives: 

PI  =  e0ii  -  r  (t'i2  +  z'22) 

=  Po  -  ri2,  (17) 

where:  t  =  total  current.     That  is,  the  power  out- 

put at  the  armature  conductors  is  the  power  input  minus  the 
i2r  loss. 

The  current  in  the  field  is: 

to  =  eb,  (18) 

hence,  the  izr  loss  in  the  field  ;  of  resistance,  n. 

to2ri  =  e262ri.  (19) 

The  hysteresis  loss  in  the  induction  motor  of  mutual  induced 
voltage,  e,  is:  e2g,  or  approximately: 

P'  =  eo29,  (20) 

in  the  synchronous  motor,  the  nominal  induced  voltage,  e,  does 
not  correspond  to  any  flux,  but  may  be  very  much  higher,  than 
corresponds  to  the  magnetic  flux,  which  gives  the  hysteresis 
loss,  as  it  includes  the  effect  of  armature  reaction,  and  the  hys- 
teresis loss  thus  is  more  nearly  represented  by  eQ2g  (20).  The 
difference,  however,  is  that  in  the  synchronous  motor  the  hys- 
teresis loss  is  supplied  by  the  mechanical  power,  and  not  the 
electric  power,  as  in  the  induction  motor. 

The  net  mechanical  output  of  the  motor  thus  is  : 

P  =  Pl  -  i02n  -  P' 


=  e0ii  -  izr  -  e26Vi  -  e2g,  (21) 

and    herefrom    follow    efficiency,    power-factor    and    apparent 
efficiency. 

44.  Considering,  as  instance,  a  typical  good  induction  motor, 
of  the  constants  : 

BO  =  500  volts; 

Fo  =  0.01  -  0.1  j; 

Z0  =  0.1  +  0.3J; 

Zi  =  0.1  +J0.3J. 


INDUCTION  MOTOR  63 

The  load  curves  of  this  motor,  as  induction  motor,  calculated 
in  the  customary  way,  are  given  in  Fig.  22. 

Converted  into  a  synchronous  motor,  it  gives  the  constants: 
Synchronous  impedance  (1) : 

Z  =  r  +  jx  =  0.1  +  10.3  j. 

Fig.  23  gives  the  load  characteristics  of  the  motor,  with  the 
power  output  as  abscissae,  with  the  direct-current  excitation, 
and  thereby  the  counter  e.m.f.,  e,  varied  with  the  load,  so  as  to 
maintain  unity  power-factor. 

The  calculation  is  made  in  tabular  form,  by  calculating  for 
various  successive  values  of  the  energy  current  (here  also  the 
total  current)  ilt  input,  the  counter  e.m.f.,  e,  by  equation  (8): 

62  =  (500  -  0.1 *,)2  +  100.61  ii2, 

the  power  input,  which  also  is  the  volt-ampere  input,  the  power- 
factor  being  unity,  is: 

Po  =  eoi'i  =  500  ii. 
From  e  follow  the  losses,  by  (17),  (19)  and  (20): 

in  armature  resistance:         0.1  i\2] 
in  field  resistance:  0.001  e2; 

hysteresis  loss:  2.5  kw. ; 

and  thus  the  power  output: 

p  =  500  ii  -  2.5  -  0.1  ti»  -  0.001  e2 

and  herefrom  the  efficiency. 

Fig.  23  gives  the  total  current  as  i,  the  nominal  induced  voltage 
as  e,  and  the  apparent  efficiency  which  here  is  the  true  efficiency, 
as  7. 

As  seen,  the  nominal  induced  voltage  has  to  be  varied  very 
greatly  with  the  load,  indeed,  almost  proportional  thereto.  That 
is,  to  maintain  unity  power-factor  in  this  motor,  the  field  excita- 
tion has  to  be  increased  almost  proportional  to  the  load. 

It  is  interesting  to  investigate  what  load  characteristics  are 
given  by  operating  at  constant  field  excitation,  that  is,  constant 
nominal  induced  voltage,  e,  as  this  would  usually  represent  the 
operating  conditions. 


64 


ELECTRICAL  APPARATUS 


FIG.  22. — Load  curves  of  standard  induction  motor. 


7 


INDUCTION   MOTOR 
DIRECT  CURRENT  EXCITATION  FOI 

UNITY  POWER  FACTOR 

e0=500  Z0  =  .1+  .3j 

Y0=.01  -A  j     Z!=.1  4-  .3j 

(Z  -.1   +  10. 3j) 

SYNCHRONOUS 


90    100  110  120  130  140   150  160  170  180  KW 


OLT 

1400] 


1300 


1200 
100 


FIG.  23. — Load  curves  at^unity  power-factor  excitation,  of  standard  induc- 
tion motor  converted  to  synchronous  motor.  J 


INDUCTION  MOTOR 


65 


Figs.  24  and  25  thus  give  the  load  characteristics  of  the  motor, 
at  constant  field  excitation,  corresponding  to: 

in  Fig.  24:  e  =  2  e0; 

in  Fig.  25:  e  =  5  eQ. 

For  different  values  of  the  energy  current,  ij,  from  zero  up  to 
the  maximum  value  possible  under  the  given  field  excitation, 


INDUCTION   MOTOR 
CONSTANT  DIRECT  CURRENT  EXCITATION 


60=500 

Zi 

(Z  =  .1  +  10.3  J* 
SYNCHRONOUS 


FIG.  24. — Load  curves  at  constant  excitation  2e,  of  standard  induction 
motor  converted  tq  synchronous  motor. 


as  given  by  equation  (12),  the  reactive  current,  iz,  is  calculated 
by  equation  (11): 

Fig.  24: 
'Fig.  25: 

The  total  current  then  is: 


i2  =  48.5  -  \/9410  -  i!2; 
iz  =  48.5  -  \/58,800  -  if. 


the  volt-ampere  input : 
the  power  input : 

5 


Q  =  e0i; 
PO  = 


66 


ELECTRICAL  APPARATUS 


the  power  output  given  by  (21),  and  herefrom  efficiency  77, 
power-factor  p  and  apparent  efficient,  7,  calculated  and  plotted. 

Figs.  24  and  25  give,  with  the  power  output  as  abscissae,  the 
total  current  input,  efficiency,  power-factor  and  apparent 
efficiency. 

As  seen  from  Figs.  24  and  25,  the  constants  of  the  motor  as 
synchronous  motor  with  constant  excitation,  are  very  bad:  the 
no-load  current  is  nearly  equal  to  full-load  current,  and  power- 


INDUCTION  MOTOR 
CONSTANT  DIRECT  CURRENT  EXCITATION 

e  =  5e 


=  .1  +.3  j 
(Z  =  .1-M0.3J) 
SYNCHRONOUS 


FIG.  25. — Load  curves  at  constant  excitation  5  e,  of  standard   induction 
motor  converted  to  synchronous  motor. 

factor  and  apparent  efficiency  are  very  low  except  in  a  narrow 
range  just  below  the  maximum  output  point,  at  which  the 
motor  drops  out  of  step. 

Thus  this  motor,  and  in  general  any  reasonably  good  induction 
motor,  would  be  spoiled  in  its  characteristics,  by  converting  it 
into  a  synchronous  motor  with  constant  field  excitation. 

In  Fig.  23  are  shown,  for  comparison,  in  dotted  lines,  the 
apparent  efficiency  taken  from  Figs.  24  and  25,  and  the  apparent 
efficiency  of  the  machine  as  induction  motor,  taken  from  Fig.  22. 


INDUCTION  MOTOR 


67 


45.  As  further  instance,  consider  the  conversion  into  a  syn- 
chronous motor  of  a  poor  induction  motor:  a  slow-speed  motor  ot 
very  high  exciting  current,  of  the  constants: 

e0  =  500; 
Fo  =  0.02  -  0.6  j; 
Z<>  =  0.1  +  0.3J; 
Zi  =  0.1  +  0.3j. 

The  load  curves  of  this  machine  as  induction  motor  are  given 
in  Fig.  20. 


LOW  SPEED  INDUCTION  MOTOR 
DIRECT  CURRENT  EXCITATION  FOR 

UNITY  POWER  FACTOR 
60  —  500  Z0=.1  -*-  .33 

Y0  =  .02-.6j       Z,  =  .1  +  .3J 
(Z   -   1  4-  2s) 
SYNCHRONOUS 


90.  .450.  .900 


.700 


10  20  30  40  50  60  70 


0  120  130  140  150  160  170  180  190  200  KW 


OUT 

1300 


55Q.110C 


500- 1000 


FIG.  26. — Load  curves  of  low-speed  high-excitation  induction  motor  con- 
verted to  synchronous  motor,  at  unity  power-factor  excitation. 

Converted  to  a  synchronous  motor,  it  has  the  constants: 
Synchronous  impedance: 

Z  =  0.1  +  1.97  j. 

Calculated  in  the  same  manner,  the  load  curves,  when  vary- 
ing the  field  excitation  with  changes  of  load  so  as  to  maintain 
unity  power-factor,  are  given  in  Fig.  26,  and  the  load  curves  for 
constant  field  excitation  giving  a  nominal  induced  voltage: 

e  =  1.5  60 
are  given  in  Fig.  27. 

As  seen,  the  increase  of  field  excitation  required  to  maintain 


68 


ELECTRICAL  APPARATUS 


unity  power-factor,  as  shown  by  curve  e  in  Fig.  26,  while  still 
considerable,  is  very  much  less  in  this  poor  induction  motor, 
than  it  was  in  the  good  induction  motor  Figs.  22  to  25. 

The  constant-excitation  load  curves,  Fig.  27,  give  character- 
istics, which  are  very  much  superior  to  those  of  the  motor  as  in- 
duction motor.  The  efficiency  is  not  materially  changed,  as  was 
to  be  expected,  but  the  power-factor,  p,  is  very  greatly  improved 
at  all  loads,  is  96  per  cent,  at  full-load,  rises  to  unity  above  full- 


LOW  SPEED  INDUCTION   MOTOR 
CONSTANT  DIRECT  CURRENT  EXCITATION 

=  1.500 

=  500  Z0=.1+.3j 

Y0=.02-.6j 

(Z=.1-K2j) 
SYNCHRONOUS 


FIG.  27. — Load  curve  of  low-speed  high-excitation  induction  motor  con- 
verted to  synchronous  motor,  at  constant  field  excitation. 

load  (assumed  as  75  kw.)  and  is  given  at  quarter-load  already 
higher  than  the  maximum  reached  by  this  machine  as  straight 
induction  motor. 

For  comparison,  in  Fig.  28  are  shown  the  curves  of  apparent 
efficiency,  with  the  power  output  as  abscissae,  of  this  slow-speed 
motor,  as: 

I  as  induction  motor  (from  Fig.  20) ; 

So  as  synchronous  motor  with  the  field  excitation  varying  to 
maintain  unity  power-factor  (from  Fig.  26) ; 

S  as  synchronous  motor  with  constant  field  excitation  (from 
Fig.  27). 


INDUCTION  MOTOR, 


69 


As  seen,  in  the  constants  at  load,  constant  excitation,  S,  is  prac- 
tically as  good  as  varying  unity  power-factor  excitation,  SQ,  drops 
below  it  only  at  partial  load,  though  even  there  it  is  very  greatly 
superior  to  the  induction-motor  characteristic,  /. 

It  thus  follows: 

By  converting  it  into  a  synchronous  motor,  by  passing  a  direct 
current  through  the  rotor,  a  good  induction  motor  is  spoiled,  but 
a  poor  induction  motor,  that  is,  one  with  very  high  exciting 
current,  is  greatly  improved. 


I     INDUCTION  MOTOR 
•     80  SYNCHRONOUS,  UNITY  POWER  FACTOR 
®     8     SYNCHRONOUS,  CONSTANT  EXCITATION 
X  CSoSYNCHR.CONCAT.INDUCT., UNITY  P.P. 
A  CS    8YNCHR.CONCAT.INDUCT., 
+  CC    COMMUTAT.MACH.CONCAT.INDUCTION 
Y    C     CONDENSER  IN  SECONDARY 


20  30  40  50  60  70  80  90  100  110  120  130  140  150  160  170 


FIG.  28. — Comparison  of  apparent  efficiency  and  speed  curves  of  high- 
excitation  induction  motor  with  various  forms  of  secondary  excitation. 

46.  The  reason  for  the  unsatisfactory  behavior  of  a  good  induc- 
tion motor,  when  operated  as  synchronous  motor,  is  found  in  the 
excessive  value  of  its  synchronous  impedance. 

Exciting  admittance  in  the  induction  motor,  and  synchronous 
impedance  in  the  synchronous  motor,  are  corresponding  quanti- 
ties, representing  the  magnetizing  action  of  the  armature  cur- 
rents. In  the  induction  motor,  in  which  the  magnetic  field  is 
produced  by  the  magnetizing  action  of  the  armature  currents, 
very  high  magnetizing  action  of  the  armature  current  is  desirable, 
so  as  to  produce  the  magnetic  field  with  as  little  magnetizing  cur- 
rent as  possible,  as  this  current  is  lagging,  and  spoils  the  power- 
factor.  In  the  synchronous  motor,  where  the  magnetic  field  is 
produced  by  the  direct  current  in  the  field  coils,  the  magnetizing 
action  of  the  armature  currents  changes  the  resultant  field  excita- 
tion, and  thus  requires  a  corresponding  change  of  the  field  current 
to  overcome  it,  and  the  higher  the  armature  reaction,  the  more 


70  ELECTRICAL  APPARATUS 

has  the  field  current  to  be  changed  with  the  load,  to  maintain 
proper  excitation.     That  is,  low  armature  reaction  is  necessary. 

In  other  words,  in  the  induction  motor,  the  armature  reaction 
magnetizes,  thus  should  be  large,  that  is,  the  synchronous  react- 
ance high  or  the  exciting  admittance  low;  in  the  synchronous 
motor  the  armature  reaction  interferes  with  the  impressed  field 
excitation,  thus  should  be  low,  that  is,  the  synchronous  imped- 
ance low  or  the  exciting  admittance  high. 

Therefore,  a  good  synchronous  motor  makes  a  poor  induction 
motor,  and  a  good  induction  motor  makes  a  poor  synchronous 
motor,  but  a  poor  induction  motor — one  of  high  exciting  admit- 
tance, as  Fig.  20 — makes  a  fairly  good  synchronous  motor. 

Here  a  misunderstanding  must  be  guarded  against:  in  the 
theory  of  the  synchronous  motor,  it  is  explained,  that  high 
synchronous  reactance  is  necessary  for  good  and  stable  synchro- 
nous-motor operation,  and  for  securing  good  power-factors  at  all 
loads,  at  constant  field  excitation.  A  synchronous  motor  of  low 
synchronous  impedance  is  liable  to  be  unstable,  tending  to  hunt 
and  give  poor  power-factors  due  to  excessive  reactive  currents. 

This  apparently  contradicts  the  conclusions  drawn  above  in 
the  comparison  of  induction  and  synchronous  motor. 

However,  the  explanation  is  found  in  the  meaning  of  high  and 
low  synchronous  reactance,  as  seen  by  expressing  the  synchro- 
nous reactance  in  per  cent. :  the  percentage  synchronous  reactance 
is  the  voltage  consumed  by  full-load  current  in  the  synchronous 
reactance,  as  percentage  of  the  terminal  voltage. 

When  discussing  synchronous  motors,  we  consider  a  synchro- 
nous reactance  of  10  to  20  per  cent,  as  low,  and  a  synchronous 
reactance  of  50  to  100  per  cent,  as  high. 

In  the  motor,  Figs.  22  to  25,  full-load  current — at  75  kw.  out- 
put— is  about  180  amp.  At  a  synchronous  reactance  of  x  = 
10.3,  this  gives  a  synchronous  reactance  voltage  at  full-load 
current,  of  1850,  or  a  synchronous  reactance  of  370  per  cent. 

In  the  poor  motor,  Figs.  20, 26  and  27,  full-load  current  is  about 
200  amp.,  the  synchronous  reactance  x  =  1.97,  thus  the  react- 
ance voltage  394,  or  79  per  cent.,  or  of  the  magnitude  of  good 
synchronous-motor  operation. 

That  is,  the  motor,  which  as  induction  motor  would  be  consid- 
ered as  of  very  high  exciting  admittance,  giving  a  low  synchro- 
nous impedance  when  converted  into  a  synchronous  motor,  would 
as  synchronous  motor,  and  from  the  viewpoint  of  synchronous- 


INDUCTION  MOTOR  71 

motor  design,  be  considered  as  a  high  synchronous  impedance 
motor,  while  the  good  induction  motor  gives  as  synchronous 
motor  a  synchronous  impedance  of  several  hundred  per  cent.,  that 
is  far  beyond  any  value  which  ever  would  be  considered  in  syn- 
chronous-motor design. 

Induction  Motor  Concatenated  with  Synchronous 

47.  Let  an  induction  machine  have  the  constants: 

F0  =  g   —  jb    =  primary  exciting  admittance, 
Z0  =  TO  +  JXQ  =  primary    self-inductive    im- 
pedance, 

Zi  =  /-]  +  jxi  =  secondary  self-inductive  im- 
pedance at  full  frequency, 
reduced  to  primary, 

and  let  the  secondary  circuit  of  this  induction  machine  be  con- 
nected to  the  armature  terminals  of  a  synchronous  machine 
mounted  on  the  induction-machine  shaft,  so  that  the  induction- 
motor  secondary  currents  traverse  the  synchronous-motor  arma- 
ture, and  let : 

Zz  =  TI  +  jxz  =  synchronous  impedance  of 
the  synchronous  machine, 
at  the  full  frequency  im- 
pressed upon  the  induction 
machine. 

The  frequency  of  the  synchronous  machine  then  is  the  fre- 
quency of  the  induction-motor  secondary,  that  is,  the  frequency 
of  the  induction-motor  slip.  The  synchronous-motor  frequency 

also  is  the  frequency  of  synchronous-motor  rotation,  or  -  times 

the  frequency  of  induction-motor  rotation,  if  the  induction  motor 
has  n  times  as  many  poles  as  the  synchronous  motor. 
Herefrom  follows: 

1  -  s 

-—      o 

or: 

*  =  ^1-  (1) 

that  is,  the  concatenated  couple  runs  at  constant  slip,  s  =  -  — r> 
thus  constant  speed, 

7? 

1  —  s  =  -      -  of  synchronism.  (2) 

n  +1 


72  ELECTRICAL  APPARATUS 

Thus  the  machine  couple  has  synchronous-motor  character- 
istics, and  runs  at  a  speed  corresponding  to  synchronous  speed 
of  a  motor  having  the  sum  of  the  induction-motor  and  syn- 
chronous-motor poles  as  number  of  poles. 

If  n  =  1,  that  is,  the  synchronous  motor  has  the  same  number 
of  poles  as  the  induction  motor, 

s  =  0.5, 
1  _  8  =  0.5, 

that  is,  the  concatenated  couple  operates  at  half  synchronous 
speed,  and  shares  approximately  equally  in  the  power  output. 

If  the  induction  motor  has  76  poles,  the  synchronous  motor 
four  poles,  n  =  19,  and: 

s  =  0.05, 
1  -  s  =  0.95, 

that  is,  the  couple  runs  at  95  per  cent,  of  the  synchronous  speed 
of  a  76-polar  machine,  thus  at  synchronous  speed  of  an  80-polar 
machine,  and  thus  can  be  substituted  for  an  80-polar  induction 
motor.  In  this  case,  the  synchronous  motor  gives  about  5 
per  cent.,  the  induction  motor  95  per  cent,  of  the  output;  the 
synchronous  motor  thus  is  a  small  machine,  which  could  be  con- 
sidered as  a  synchronous  exciter  of  the  induction  machine. 
48.  Let: 

$o  =  e'o  +  J0"o  —  voltage  impressed  upon  in- 
duction motor. 

EI  =  e'\  +  je"\  =  voltage  induced  in  induc- 
tion motor,  by  mutual 
magnetic  flux,  reduced  to 
full  frequency. 

Ez  =  e'z  -\-  je"z  =  nominal  induced  voltage 
of  synchronous  motor,  re- 
duced to  full  frequency. 

70  =  i'Q  —  ji"0  =  primary  current  in  induc- 

tion motor. 

71  =  i'i  —  ji"\  =  secondary    current    of    in- 

duction   motor    and   cur- 
rent in  synchronous  motor. 

Denoting  by  Z*  the  impedance,  Z,  at  frequency,  s,  it  is: 
Total  impedance  of  secondary  circuit,  at  frequency,  s: 
Z*  =  Zf  +  Z2S 

=  (ri  +  ra)  -f  /(zi  +  z2),  (3) 


INDUCTION  MOTOR  73 

and  the  equations  are: 
in  primary  circuit: 

E0  =  E}  +  Zo/o;  (4) 

in  secondary  circuit: 

sE,  *=  sE2  +  Z*/i;  (5) 

and,  current: 

/o  =  /,  +  YE,.  (6) 

From  (6)  follows: 

/!  =  70  -  F#i,  (7) 

and,  substituting  (7)  into  (5): 


hence: 

sE2  + 


substituting  (8)  into  (4)  gives: 

-r,        sE^  ~h  (Z8  -}-  sZo 


s  +  Z*? 
and,  transposed: 

Z'A  .  rZ 


or: 


(l  +  7  F)  -  ^  +[      +  Z.(l  +  T*  F)]  /»,          (9) 


Denoting : 

s 

(ID 


and: 


it  is,  substituting  into  (9)  and  (10)  : 

#o  (1  +  Z'Y)  =  #2  +  (Z'  +  ZQ  +  Z%>7)  /o,  (12) 


74  ELECTRICAL  APPARATUS 

Denoting : 

1  +  *ZY  =  $  =  e'  +  je"  (14) 

as  a  voltage  which  is  proportional  to  the  nominal  induced  voltage 
of  the  synchronous  motor,  and : 

Zf 

Y+-&Y  +  Zo  =  z  =  r  +  jx  (15) 

and  substituting  (14)  and  (15)  into  (13),  gives: 

E  =  E0  -  Z/o.  (16) 

This  is  the  standard  synchronous-motor  equation,  with  im- 
pressed voltage,  E0}  current,  70,  synchronous  impedance,  Z,  and 
nominal  induced  voltage,  E. 

Choosing  the  impressed  voltage,  EQ  =  eQ  as  base  line,  and 
substituting  into  (16),  gives: 

e'  +  je"  =  (eQ  -  ri'o  -  xi"Q)  -  j  (xi'0  -  n"0),          (17) 
and,  absolute: 


From  this  equation  (18)  the  load  and  speed  curves  of  the 
concatenated  couple  can  now  be  calculated  in  the  same  manner 
as  in  any  synchronous  motor. 

That  is,  the  concatenated  couple,  of  induction  and  synchronous 
motor,  can  be  replaced  by  an  equivalent  synchronous  motor  of 
the  constants,  e,  €Q,  Z  and  /o. 

49.  The  power  output  of  the  synchronous  machine  is : 

p2  =  //„  *&/', 

where : 

/a+jb,   c+jd/' 

denotes  the  effective  component  of  the  double-frequency  prod- 
uct:   (ac  +  bd)',   see   ''Theory  and  Calculation  of  Alternating- 
current  Phenomena,"  Chapter  XVI,  5th  edition. 
The  power  output  of  the  induction  machine  is : 

Px  =  //!,  (1  -  s)  E,/',  .    (20) 

thus,  the  total  power  output  of  the  concatenated  couple : 

P  =  Pi  +  P* 

=  //i,  sE2  +  (1  -  «)#!/';  (21) 


INDUCTION  MOTOR  75 

substituting  (7)  into  (21) : 

P  =  /h-  YEly  sE2  +  (1  -  s)#i/';  (22) 

from  (8)  follows: 

sE2  =  $1(8+-Z'Y)  -  Z°IQ, 
and  substituting  this  into  (22),  gives: 

P  =  //o  -  F#i,  tfi  (1  +  Z*F)  -  Z'/o/';  (23) 

from  (4)  follows: 

EI    =   EQ   —   ZQ!Q, 

and  substituting  this  into  (23)  gives: 

P  =  //o  (1  +  Z0F)  -  YE0}  #0  (1  +  Z'Y)  - 

IQ  (Z5  +  Z0  +  ZoZ-7)/'.     (24) 

Equation  (24)  gives  the  power  output,  as  function  of  impressed 
voltage,  EQ,  and  supply  current,  /o. 

The  power  input  into  the  concatenated  couple  is  given  by: 

Po  =  /EQ)  /o/',  (25) 

or,  choosing  EQ  =  eQ  as  base  line: 

Po  =  coi'o.  (26) 

The  apparent  power,  or  volt-ampere  input  is  given  by: 

Q  =  eQio,  (27) 

where : 

to  =  ViVTW 

is  the  total  primary  current. 

From  P,  Po  and  Q  now  follow  efficiency,  power-factor  and 
apparent  efficiency. 

50.  As  an  instance  may  be  considered  the  power-factor  control 
of  the  slow-speed  80-polar  induction  motor  of  Fig.  20,  by  a  small 
synchronous  motor  concatenated  into  its  secondary  circuit. 

Impressed  voltage: 

eQ  =  500  volts. 

Choosing  a  four-polar  synchronous  motor,  the  induction 
machine  would  have  to  be  redesigned  with  76  poles,  giving: 

n  =      19, 
s  =  0.05. 


76  ELECTRICAL  APPARATUS 

With  the  same  rotor  diameter  of  the  induction  machine,  the 
pole  pitch  would  be  increased  inverse  proportional  to  the  number 
of  poles,  and  the  exciting  susceptance  decreased  with  the  square 
thereof,  thus  giving  the  constants : 

Yo  =  g   -jb    =  0.02  -  0.54  j; 
Z0  =  r0+jx0  =  0.1  +  0.3J; 
Zi  =  n  +  jx!  =  0.1  +  0.3  j. 

Assuming  as  synchronous  motor  synchronous  impedance, 
reduced  to  full  frequency: 

Z2  =  r2+jxz  =  0.02  + 0.2  j 
this  gives,  for  s  =  0.05: 

Z*  =  (n  +  r2)  +  js  fa  +  x2)  =  0.12  +  0.025  j, 
and: 

Z'  =  r'  +jx'  =  ^  =  2.4  +  0.5  j, 

o 

Z    =  r+  jx  =  0.84  +  1.4  j, 
and  from  (14) : 

* 


1.32-  1.29 /' 

«"iSr  1 11  , V  :.'...,., 

thus: 

=  (500  -  0.84 1'0  -  1.4 1"0)2  +  (1.4  i'Q  -  0.84  To)2,       (28) 


and  the  power  output : 

p  =  /70(0.836  +  0.048  j)  -  (10  -  270  j), 

(508  -  32  j)  -  /o  (0.241  +  0.326  j}/'.        (29) 

51.  Fig.  29  shows  the  load  curves  of  the  concatenated  couple, 
under  the  condition  that  the  synchronous-motor  excitation  and 
thus  its  nominal  induced  voltage,  62,  is  varied  so  as  to  maintain 
unity  power-factor  at  all  loads,  that  is: 

i"o  =  0; 
this  gives  from  equation  (28) : 

^  =  (500  -  0.84 i'0)2  +  1.96 iV, 


INDUCTION  MOTOR 


77 


LOW  SPEED  INDUCTION  MOTOR 

WITH  LOW   FREQUENCY  SYNCHRONOUS  IN  SECONDARY, 
EXCITED  FOR  UNITY  POWER  FACTOR 
00  =  500  Z0=.1-*-.3j 

Y0  = -02 -.54:?         Z1  =  .1-*-.3j 
5    =  .05  Z2  =  .02  +.2J 

SYNCHRONOUS 


90  100  110  120  130  140  150  160  170  180  190  KW 


AMPS 

600-J1200 


550. 1100 
500. 1000 


OLTJ 

1400 


1300 


FIG.  29. — Load  curves  of  high-excitation  induction  motor  concatenated  with 
synchronous,  at  unity  power-factor  excitation. 


WITH   LOW 
C 

LOW  SPE 
FREQUElv 
3NSTANT 

e0=  500 

Y0--02 
S  =  .05 
S 

ED  1 
ICY 
EXC 

NDUCTION   MOTOF 
SYNCHRONOUS  IN 
IITATION,    e  =  1.3 
Z0=  .1  4-  .33 
Vj     Zt=..H-.3j. 
Z2=  .02-1-.  2J 
HRONOUS 

? 

SECONDARY 
'«• 

YNC 

% 

100 

AMPS, 

_450. 
_400 
-350 
_300 
_250. 
_200. 
_150. 
_100 
_50. 

/ 

P 

'j 

) 

90 

\ 

80 

7 

70 

7 

f 

^ 

/ 

X 

60 

/ 

^ 

^ 

"!" 

50 

f 

^.x- 

X 

40 

/ 

t^ 

^ 

^ 

30 

7 

^^ 

^ 

^^^ 

20 

/ 

^ 

^ 

10 

/I 

0      2 

0      3 

0      4 

3       £ 

0      6 

0      7 

0       8 

I)       9 

0      11 

0     1 

0    l: 

JO    IS 

^)    KW 

FIG.  30. — Load  curves  of  high-excitation  induction  motor  concatenated  with 
synchronous,  at  constant  excitation. 


78  ELECTRICAL  APPARATUS 

p  =  /(0.8361  i'0  -  10)  +  j  (0.048  i'Q  +  270), 

(508  =  0.241  t'o)  -  j  (32  +  0.326  i'0/' 
=  (0.836  t'o  -  10)  (508  -  0.241  i'0)  -  (0.048 *'„  +  270) 
(32  +  0.326  i'o). 

As  seen  from  the  curve,  62,  of  the  nominal  induced  voltage,  the 
synchronous  motor  has  to  be  overexcited  at  all  loads.  However, 
ez  first  decreases,  reaches  a  minimum  and  then  increases  again, 
thus  is  fairly  constant  over  a  wide  range  of  load,  so  that  with 
this  type  of  motor,  constant  excitation  should  give  good  results. 

Fig.  30  then  shows  the  load  curves  of  the  concatenated  couple 
for  constant  excitation,  on  overexcitation  of  the  synchronous 
motor  of  70  per  cent.,  or 

e2  =  850  volts. 

(It  must  be  kept  in  mind,  that  e2  is  the  voltage  reduced  to  full 
frequency  and  turn  ratio  1 : 1  in  the  induction  machine :  At  the 
slip,  s  =  0.05,  the  actual  voltage  of  the  synchronous  motor  would 
be  se2  =  42.5  volts,  even  if  the  number  of  secondary  turns  of  the 
induction  motor  equals  that  of  the  primary  turns,  and  if,  as 
usual,  the  induction  motor  is  wound  for  less  turns  in  the  secondary 
than  in  the  primary,  the  actual  voltage  at  the  synchronous  motor 
terminals  is  still  lower.) 

As  seen  from  Fig.  30: 

the  power-factor  is  practically  unity  over  the  entire  range  of 
load,  from  less  than  one-tenth  load  up  to  the  maximum  output 
point,  and  the  current  input  into  the  motor  thus  is  practically 
proportional  to  the  load. 

The  load  curves  of  this  concatenated  couple  thus  are  superior 
to  those,  which  can  be  produced  in  a  synchronous  motor  at  con- 
stant excitation. 

For  comparison,  the  curve  of  apparent  efficiency,  from  Fig.  30, 
is  plotted  as  CS  in  Fig.  28.  It  merges  indistinguishably  into  the 
unity  power-factor  curve,  S0,  except  at  its  maximum  output 
point. 

Induction  Motor  Concatenated  with  Commutating 
Machine 

52.  While  the  alternating-current  commutating  machine,  espe- 
cially of  the  polyphase  type,  is  rather  poor  at  higher  frequencies, 
it  becomes  better  at  lower  frequencies,  and  at  the  extremely  low 
frequency  of  the  induction-motor  secondary,  it  is  practically  as 


INDUCTION  MOTOR  79 

good  as  the  direct-current  commutating  machine,  and  thus  can 
be  used  to  insert  low-frequency  voltage  into  the  induction-motor 
secondary. 

With  series  excitation,  the  voltage  of  the  commutating  machine 
is  approximately  proportional  to  the  secondary  current,  and  the 
speed  characteristic  of  the  induction  motor  remains  essentially 
the  same :  a  speed  decreasing  from  synchronism  at  no-load,  by  a 
slip,  s,  which  increases  with  the  load. 

With  shunt  excitation,  the  voltage  of  the  commutating  machine 
is  approximately  constant,  and  the  concatenated  couple  thus 
tends  toward  a  speed  differing  from  synchronism. 

In  either  case,  however,  the  slip,  s,  is  not  constant  and  independ- 
ent of  the  load,  and  the  motor  couple  not  synchronous,  as  when 
using  a  synchronous  machine  as  second  motor,  but  the  motor 
couple  is  asynchronous,  decreasing  in  speed  with  increase  of  load. 

The  phase  relation  of  the  voltage  produced  by  the  commutating 
machine,  with  regards  to  the  secondary  current  which  traverses 
it,  depends  on  the  relation  of  the  commutator  brush  position 
with  regards  to  the  field  excitation  of  the  respective  phases,  and 
thereby  can  be  made  anything  between  0  and  2  IT,  that  is,  the 
voltage  inserted  by  the  commutating  machine  can  be  energy 
voltage  in  phase — reducing  the  speed — or  in  opposition  to  the 
induction-motor  induced  voltage — increasing  the  speed;  or  it 
may  be  a  reactive  voltage,  lagging  and  thereby  supplying  the 
induction-motor  magnetizing  current,  or  leading  and  thereby 
still  further  lowering  the  power-factor.  Or  the  commutating 
machine  voltage  may  be  partly  in  phase — modifying  the  speed— 
and  partly  in  quadrature — modifying  the  power-factor. 

Thus  the  commutating  machine  in  the  induction-motor 
secondary  can  be  used  for  power-factor  control  or  for  speed 
control  or  for  both. 

It  is  interesting  to  note  that  the  use  of  the  commutating  ma- 
chine in  the  induction- mot  or  secondary  gives  two  independent 
variables:  the  value  of  the  voltage,  and  its  phase  relation  to  the 
current  of  its  circuit,  and  the  motor  couple  thus  has  two  degrees 
of  freedom.  With  the  use  of  a  synchronous  machine  in  the 
induction-motor  secondary  this  is  not  the  case;  only  the  voltage 
of  the  synchronous  machine  can  be  controlled,  but  its  phase 
adjusts  itself  to  the  phase  relation  of  the  secondary  circuit,  and 
the  synchronous-motor  couple  thus  has  only  one  degree  of  free- 
dom. The  reason  is :  with  a  synchronous  motor  concatenated  to 


80  ELECTRICAL  APPARATUS 

the  induction  machine,  the  phase  of  the  synchronous  machine  is 
fixed  in  space,  by  the  synchronous-motor  poles,  thus  has  a  fixed 
relation  with  regards  to  the  induction-motor  primary  system. 
As,  however,  the  induction-motor  secondary  has  no  fixed  position 
relation  with  regards  to  the  primary,  but  can  have  any  position 
slip,  the  synchronous-motor  voltage  has  no  fixed  position  with 
regards  to  the  induction-motor  secondary  voltage  and  current, 
thus  can  assume  any  position,  depending  on  the  relation  in  the 
secondary  circuit.  Thus  if  we  assume  that  the  synchronous- 
motor  field  were  shifted  in  space  by  a  position  degrees  (electrical) : 
this  would  shift  the  phase  of  the  synchronous-motor  voltage  by 
a  degrees,  and  the  induction-motor  secondary  would  slip  in  posi- 
tion by  the  same  angle,  thus  keep  the  same  phase  relation  with 
regards  to  the  synchronous-motor  voltage.  In  the  couple  with 
a  commutating  machine  as  secondary  motor,  however,  the  posi- 
tion of  the  brushes  fixes  the  relation  between  commutating- 
machine  voltage  and  secondary  current,  and  thereby  imposes  a 
definite  phase  relation  in  the  secondary  circuit,  irrespective  of 
the  relations  between  secondary  and  primary,  and  no  change  of 
relative  position  between  primary  and  secondary  can  change  this 
phase  relation  of  the  commutating  machine. 

Thus  the  commutating  machine  in  the  secondary  of  the  induc- 
tion machine  permits  a  far  greater  variation  of  conditions  of 
operation,  and  thereby  gives  a  far  greater  variety  of  speed  and 
load  curves  of  such  concatenated  couple,  than  is  given  by  the 
use  of  a  synchronous  motor  in  the  induction-motor  secondary. 

53.  Assuming  the  polyphase  low-frequency  commutating 
machine  is  series-excited,  that  is,  the  field  coils  (and  compensat- 
ing coils,  where  used)  in  series  with  the  armature.  Assuming 
also  that  magnetic  saturation  is  not  reached  within  the  range  of 
its  use. 

The  induced  voltage  of  the  commutating  machine  then  is 
proportional  to  the  secondary  current  and  to  the  speed. 

Thus:  ez  =  pii  (1) 

is  the  commutating-machine  voltage  at  full  synchronous  speed, 
where  i\  is  the  secondary  current  and  p  a  constant  depending 
on  the  design. 

At  the  slip,  s,  and  thus  the  speed  (1  —  s),  the  commutating 
machine  voltage  thus  is: 

(1  -  s)  e2  =  (1  -  s)  pi,.  (2) 


INDUCTION  MOTOR  81 

As  this  voltage  may  have  any  phase  relation  with  regards  to 
the  current,  ii,  we  can  put: 

#2  =  (pi  +  jpa)/i  (3) 

where  : 


P  =  Vpi2  +  p22  (4) 

and: 

tan  co  =  ^  (5) 

Pi 

is  the  angle  of  brush  shift  of  the  commutating  machine. 

(pi  +  jp2)  is  of  the  nature  and  dimension  of  an  impedance, 
and  we  thus  can  put  : 

Z°  =  pi  +  jp2  (6) 

as  the  effective  impedance  representing  the  commutating  machine. 

At  the  speed  (1  —  s), 

the    commutating    machine    is    represented    by    the    effective 

impedance: 

(1  -  s)  Z°  =  (1  -  s)  Pl  +  j  (1  -  s)  p2.  (7) 

It  must  be  understood,  however,  that  in  the  effective  impedance 
of  the  commutating  machine, 

Z°    =    Pl    +  JP2, 

Pi  as  well  as  p2  may  be  negative  as  well  as  positive. 

That  is,  the  energy  component  of  the  effective  impedance,  or 
the  effective  resistance,  pi,  of  the  commutating  machine,  may  be 
negative,  representing  power  supply.  This  simply  means,  that 
the  commutator  brushes  are  set  so  as  to  make  the  commutating 
machine  an  electric  generator,  while  it  is  a  motor,  if  pi  is  positive. 

If  pi  =  0,  the  commutating  machine  is  a  producer  of  wattless 
or  reactive  power,  inductive  for  positive,  anti-inductive  for 
negative,  p%. 

The  calculation  of  an  induction  motor  concatenated  with  a 
commutating  machine  thus  becomes  identical  with  that  of  the 
straight  induction  motor  with  short-circuited  secondary,  except 
that  in  place  of  the  secondary  inductive  impedance  of  the  induc- 
tion motor  is  substituted  the  total  impedance  of  the  secondary 
circuit,  consisting  of: 

1.  The  secondary  self-inductive  impedance  of  the  induction 
machine. 


82  ELECTRICAL  APPARATUS 

2.  The  self-inductive  impedance  of  the  commutating  machine 
comprising  resistance  and  reactance  of  armature  and  of  field, 
and  compensating  winding,  where  such  exists. 

3.  The   effective   impedance   representing   the    commutating 
machine. 

It  must  be  considered,  however,  that  in  (1)  and  (2)  the  re- 
sistance is  constant,  the  reactance  proportional  to  the  slip,  s, 
while  (3)  is  proportional  to  the  speed  (1  —  s). 
54.  Let: 

YQ  =  g  —  jb  =  primary  exciting  admittance 
of  the  induction  motor. 

Zo  =  r0  -f  jxo  =  primary  self-inductive  im- 
pedance of  the  induction 
motor. 

Zi  =  7*1  +  jxi  =  secondary  self-inductive  im- 
pedance of  the  induction 
motor,  reduced  to  full 
frequency. 

Z2  =  r2  +  jx-2  —  self-inductive  impedance  of 
the  commutating  machine, 
reduced  to  full  frequency. 

£o  _  pl  _|_  jp2  =  effective,  impedance  repre- 
senting the  voltage  in- 
duced in  the  commutating 
machine,  reduced  to  full 
frequency. 

The  total  secondary  impedance,  at  slip,  s,  then  is  : 


Z*  =  (n  +  jsxi)  +  (r2  +  jsxz)  -t-  (1  -  s)  (pl  +  jpd 

=  [r,  +  r2  +  (1  ->)  Pl]  +  j[s  (x,  +  a?2)  +  (1  ~  s)  pz]   (8) 

and,  if  the  mutual  inductive  voltage  of  the  induction  motor  is 
chosen  as  base  line,  e,  in  the  customary  manner, 
the  secondary  current  is: 

/i  =  T     =  (ai—  jaz)e,  (9) 


where : 

-    g  frl   +  r2  +   (1    -  S)  Pl] 

m 

=  s  [s  (xi  +  xz)  +  (1  -  s)  pz] 
m 


(10) 


INDUCTION  MOTOR 


83 


and: 


m 


-  s) 


The  remaining  calculation  is  the  same  as  on  page  318  of 
"  Theoretical  Elements  of  Electrical  Engineering,"  4th  edition. 

As  an  instance,  consider  the  concatenation  of  a  low-frequency 
commutating  machine  to  the  low-speed  induction  motor,  Fig.  20. 

The  constants  then  are: 


Impressed  voltage: 
Exciting  admittance: 
Impedances : 


e0  =  500; ' 
Fo  =  0.02  -  0.6  j; 
Zo  =  0.1  +  0.3J; 
Zx  =  0.1  +0.3.?; 
Z2  =  0.02  + 0.3  j; 
Z°  =  -  0.2  j. 


LOW  SPEED  INDUCTION   MOTOR 

WITH   LOW  FREQUENCY  COMMUTATING  MACHINE  IN  SECONDARY, 

SERIES  EXCITED  FOR  ANTI-INDUCTIVE  REACTIVE  VOLTAGE 


Z2=.02  +  . 
ASYNCHRONOUS 


10  20  30  40  50  60  70  80 


100  110  120  130  140  1ft)  160  170  180  190  *v 


FIG.  31. — Load  curves  of  high-excitation  induction  motor  concatenated  with 
commutating  machine  as  reactive  anti-inductive  impedance. 

That  is,  the  commutating  machine  is  adjusted  to  give  only 
reactive  lagging  voltage,  for  power-factor  compensation. 
It  then  is : 

Z°  =  0.12  +  j  [0.6  s  -  0.2  (1  -  «)]. 

The  load  curves  of  this  motor  couple  are  shown  in  Fig.  31.     As 


84  ELECTRICAL  APPARATUS 

seen,  power-factor  and  apparent  efficiency  rise  to  high  values,  and 
even  the  efficiency  is  higher  than  in  the  straight  induction  motor. 
However,  at  light-load  the  power-factor  and  thus  the  apparent 
efficiency  falls  off,  very  much  in  the  same  manner  as  in  the  con- 
catenation with  a  synchronous  motor. 

It  is  interesting  to  note  the  relatively  great  drop  of  speed  at 
light-load,  while  at  heavier  load  the  speed  remains  more  nearly 
constant.  This  is  a  general  characteristic  of  anti-inductive  im- 
pedance in  the  induction-motor  secondary,  and  shared  by  the 
use  of  an  electrostatic  condenser  in  the  secondary. 

For  comparison,  on  Fig.  28  the  curve  of  apparent  efficiency  of 
this  motor  couple  is  shown  as  CC. 

Induction  Motor  with  Condenser  in  Secondary  Circuit 

55.  As  a  condenser  consumes  leading,  that  is,  produces  lagging 
reactive  current,  it  can  be  used  to  supply  the  lagging  component 
of  current  of  the  induction  motor  and  thereby  improve  the 
power-factor. 

Shunted  across  the  motor  terminals,  the  condenser  consumes  a 
constant  current,  at  constant  impressed  voltage  and  frequency, 
and  as  the  lagging  component  of  induction-motor  current  in- 
creases with  the  load,  the  characteristics  of  the  combination  of 
motor  and  shunted  condenser  thus  change  from  leading  current 
at  no-load,  over  unity  power-factor  to  lagging  current  at  overload. 
As  the  condenser  is  an  external  apparatus,  the  characteristics  of 
the  induction  motor  proper  obviously  are  not  changed  by  a 
shunted  condenser. 

As  illustration  is  shown,  in  Fig.  32,  the  slow-speed  induction 
motor  Fig.  20,  shunted  by  a  condenser  of  125  kva.  per  phase. 
Fig.  32  gives  efficiency,  77,  power-factor,  p,  and  apparent  efficiency, 
7,  of  the  combination  of  motor  and  condenser,  assuming  an 
efficiency  of  the  condenser  of  99.5  per  cent.,  that  is,  0.5  per  cent, 
loss  in  the  condenser,  or  Z  =  0.0025  —  0.5j,  that  is,  a  condenser 
just  neutralizing  the  magnetizing  current. 

However,  when  using  a  condenser  in  shunt,  it  must  be  realized 
that  the  current  consumed  by  the  condenser  is  proportional  to  the 
frequency,  and  therefore,  if  the  wave  of  impressed  voltage  is 
greatly  distorted,  that  is,  contains  considerable  higher  harmonics 
— especially  harmonics  of  high  order — the  condenser  may  produce 
considerable  higher-frequency  currents,  and  thus  by  distortion 


INDUCTION  MOTOR 


85 


of  the  current  wave  lower  the  power-factor,  so  that  in  extreme 
cases  the  shunted  condenser  may  actually  lower  the  power- 
factor.  However,  with  the  usual  commercial  voltage  wave 
shapes,  this  is  rarely  to  be  expected. 

In  single-phase  induction  motors,  the  condenser  may  be  used 
in  a  tertiary  circuit,  that  is,  a  circuit  located  on  the  same  member 
(usually  the  stator)  as  the  primary  circuit,  but  displaced  in  posi- 


LOW  SPEED  INDUCTION   MOTOR 
WITH  SHUNTED  CONDENSER 
6o=500  Z0 


30      40       50      6,0      70       80      90      100     110 


FIG.  32. — Load  curves  of  high-excitation  induction  motor  with  shunted 

condenser. 

tion  therefrom,  and  energized  by  induction  from  the  secondary. 
By  locating  the  tertiary  circuit  in  mutual  induction  also  with  the 
primary,  it  can  be  used  for  starting  the  single-phase  motor,  and 
is  more  fully  discussed  in  Chapter  V. 

A  condenser  may  also  be  used  in  the  secondary  of  the  induction 
motor.  That  is,  the  secondary  circuit  is  closed  through  a  con- 
denser in  each  phase.  As  the  current  consumed  by  a  condenser  is 
proportional  to  the  frequency,  and  the  frequency  in  the  secondary 
circuit  varies,  decreasing  toward  zero  at  synchronism,  the  cur- 
rent consumed  by  the  condenser,  and  thus  the  secondary  current 
of  the  motor  tends  toward  zero  when  approaching  synchronism, 


86  ELECTRICAL  APPARATUS 

and  peculiar  speed  characteristics  result  herefrom  in  such  a 
motor.  At  a  certain  slip,  s,  the  condenser  current  just  balances 
all  the  reactive  lagging  currents  of  the  induction  motor,  resonance 
may  thus  be  said  to  exist,  and  a  very  large  current  flows  into  the 
motor,  and  correspondingly  large  power  is  produced.  Above  this 
"  resonance  speed,"  however,  the  current  and  thus  the  power 
rapidly  fall  off,  and  so  also  below  the  resonance  speed. 

It  must  be  realized,  however,  that  the  frequency  of  the  sec- 
ondary is  the  frequency  of  slip,  and  is  very  low  at  speed,  thus  a 
very  great  condenser  capacity  is  required,  far  greater  than  would 
be  sufficient  for  compensation  by  shunting  the  condenser  across 
the  primary  terminals.  In  view  of  the  low  frequency  and  low 
voltage  of  the  secondary  circuit,  the  electrostatic  condenser 
generally  is  at  a  disadvantage  for  this  use,  but  the  electrolytic 
condenser,  that  is,  the  polarization  cell,  appears  better  adapted. 

56.  Let  then,  in  an  induction  motor,  of  impressed  voltage,  €Q: 

YQ  =  g   —  jb     =  exciting  admittance; 

Z0  =  TQ  +  jxo  =  primary  self-inductive  impe- 

dance; 
Zi  =  TI  -f  jxi  =  secondary  self-inductive  im- 

pedance at  full  frequency; 

and  let  the  secondary  circuit  be  closed  through  a  condenser  of 
capacity  reactance,  at  full  frequency: 

Z2  =  TI  —  jx2, 

where  r2,  representing  the  energy  loss  in  the  condenser,  usually  is 
very  small  and  can  be  neglected  in  the  electrostatic  condenser, 
so  that  : 


The  inductive  reactance,  Xi,  is  proportional  to  the  frequency, 
that  is,  the  slip,  s,  and  the  capacity  reactance,  x2,  inverse  propor- 
tional thereto,  and  the  total  impedance  of  the  secondary  circuit, 
at  slip,  s,  thus  is  : 

Z'  =  r1+j(sxl-^),  (1) 

thus  the  secondary  current  : 

T        es 
****& 

=  e(ai-  ja2),  (2) 


INDUCTION  MOTOR  87 

where : 

m 


(3) 


All  the  further  calculations  of  the  motor  characteristics  now 
are  the  same  as  in  the  straight  induction  motor. 

As  instance  is  shown  the  low-speed  motor,  Fig.  20,  of  constants : 

e0  =  500; 
Fo  =  0.02  -  0.6  j; 
Zo  =  0.1  -fO.3.?; 
Zl  =  0.1  +  0.3  j; 

with  the  secondary  closed  by  a  condenser  of  capacity  impedance : 

Z2  =  -  0.012  j, 
thus  giving: 

Z*  =  0.1  +  0.3 

Fig.  33  shows  the  load  curves  of  this  motor  with  condenser 
in  the  secondary.  As  seen,  power-factor  and  apparent  effi- 
ciency are  high  at  load,  but  fall  off  at  light-load,  being  similar 
in  character  as  with  a  commutating  machine  concatenated  to 
the  induction  machine,  or  with  the  secondary  excited  by  direct 
current,  that  is,  with  conversion  of  the  induction  into  a  synchro- 
nous motor. 

Interesting  is  the  speed  characteristic:  at  very  light-load  the 
speed  drops  off  rapidly,  but  then  remains  nearly  stationary  over 
a  wide  range  of  load,  at  10  per  cent.  slip.  It  may  thus  be  said, 
that  the  motor  tends  to  run  at  a  nearly  constant  speed  of  90  per 
cent,  of  synchronous  speed. 

The  apparent  efficiency  of  this  motor  combination  is  plotted 
once  more  in  Fig.  28,  for  comparison  with  those  of  the  other 
motors,  and  marked  by  C. 

Different  values  of  secondary  capacity  give  different  operating 
speeds  of  the  motor:  a  lower  capacity,  that  is,  higher  capacity 


88 


ELECTRICAL  APPARATUS 


reactance,  x2,  gives  a  greater  slip,  s,  that  is,  lower  operating 
speed,  and  inversely,  as  was  discussed  in  Chapter  I. 

57.  It  is  interesting  to  compare,  in  Fig.  28,  the  various  methods 
of  secondary  excitation  of  the  induction  motor,  in  their  effect  in 
improving  the  power-factor  and  thus  the  apparent  efficiency  of 
a  motor  of  high  exciting  current  and  thus  low  power-factor,  such 
as  a  slow-speed  motor. 

The  apparent  efficiency  characteristics  fall  into  three  groups: 


LOW  SPEED  INDUCTION  MOTOR 
WITH  CONDENSER  IN  SECONDARY  CIRCUIT 

=  500 
Y=.02  -.6j  Z,=.1  +.3J 

Z2   =      -.012J 
ASYNCHRONOUS 


90     100    110    120   130    140    150    160    170    180  KW 


10     20     30      40     50     60     70 


FIG.  33. — Load  curves  of  high-excitation  induction  motor  with  condensers  in 

secondary  circuits. 


1.  Low  apparent  efficiency  at  all  loads:  the  straight  slow- 
speed  induction  motor,  marked  by  7. 

2.  High  apparent  efficiency  at  all  loads : 

The  synchronous  motor  with  unity  power^factor  excitation,  So. 

Concatenation  to  synchronous  motor  with  unity  power-factor 
excitation,  CSQ. 

Concatenation  to  synchronous  motor  with  constant  excitation, 
CS. 

These  three  curves  are  practically  identical,  except  at  great 
overloads. 

3.  Low    apparent    efficiency    at    light-loads,    high    apparent 


INDUCTION  MOTOR  89 

efficiency  at  load,  that  is,  curves  starting  from  (1)  and  rising  up 
to  (2). 

Hereto  belong:     The  synchronous  motor  at  constant  excita- 
tion, marked  by  S. 

Concatenation  to  a  commutating  machine, 
CC. 

Induction  motor  with  condenser  in  secondary 
circuit,  C. 

These  three  curves  are  very  similar,  the  points  calculated  for 
the  three  different  motor  types  falling  within  the  narrow  range 
between  the  two  limit  curves  drawn  in  Fig.  28. 

Regarding  the  speed  characteristics,  two  types  exist :  the  motors 
So,  S,  CSo  and  CS  are  synchronous,  the  motors  7,  CC  and  C  are 
asynchronous. 

In  their  efficiencies,  there  is  little  difference  between  the 
different  motors,  as  is  to  be  expected,  and  the  efficiency  curves 
are  almost  the  same  up  to  the  overloads  where  the  motor  begins 
to  drop  out  of  step,  and  the  efficiency  thus  decreases. 

Induction  Motor  with  Commutator 

58.  Let,  in  an  induction  motor,  the  turns  of  the  secondary 
winding  be  brought  out  to  a  commutator.  Then  by  means  of 
brushes  bearing  on  this  commutator,  currents  can  be  sent  into 
the  secondary  winding  from  an  outside  source  of  voltage. 

Let  then,  in  Fig.  34,  the  full-frequency  three-phase  currents 
supplied  to  the  three  commutator  brushes  of  such  a  motor  be 
shown  as  A.  The  current  in  a  secondary  coil  of  the  motor, 
supplied  from  the  currents,  A,  through  the  commutator,  then  is 
shown  asB.  Fig.  34  corresponds  to  a  slip,  s  =  J£.  As  seen  from 
Fig.  34,  the  commutated  three-phase  current,  B,  gives  a  resultant 
effect,  which  is  a  low-frequency  wave,  shown  dotted  in  Fig.  34 
B,  and  which  has  the  frequency  of  slip,  s,  or,  in  other  words,  the 
commutated  current,  B,  can  be  resolved  into  a  current  of  fre- 
quency, s,  and  a  higher  harmonic  of  irregular  wave  shape. 

Thus,  the  effect  of  low-frequency  currents,  of  the  frequency 
of  slip,  can  be  produced  in  the  induction-motor  secondary  by 
impressing  full  frequency  upon  it  through  commutator  and 
brushes. 

The  secondary  circuit,  through  commutator  and  brushes,  can 
be  connected  to  the  supply  source  either  in  series  to  the  primary, 


90 


ELECTRICAL  APPARATUS 


or  in  shunt  thereto,  and  thus  gives  series-motor  characteristics, 
or  shunt-motor  characteristics. 

In  either  case,  two  independent  variables  exist,  the  value  of 
the  voltage  impressed  upon  the  commutator,  and  its  phase, 
and  the  phase  of  the  voltage  supplied  to  the  secondary  circuit 
may  be  varied,  either  by  varying  the  phase  of  the  impressed 
voltage  by  a  suitable  transformer,  or  by  shifting  the  brushes  on 
the  commutator  and  thereby  the  relative  position  of  the  brushes 
with  regards  to  the  stator,  which  has  the  same  effect. 

However,  with  such  a  commutator  motor,  while  the  resultant 
magnetic  effect  of  the  secondary  currents  is  of  the  low  frequency 


FIG.  34. — Commutated  full-frequency  current  in  induction  motor 
secondary. 

of  slip,  the  actual  current  in  each  secondary  coil  is  of  full  fre- 
quency, as  a  section  or  piece  of  a  full-frequency  wave,  and  thus 
it  meets  in  the  secondary  the  full-frequency  reactance.  That  is, 
the  secondary  reactance  at  slip,  s,  is  not :  Zs  =  TI  +  jsxi,  but  is : 
Z*  =  ri  -f  jxi,  in  other  words  is  very  much  larger  than  in  the 
motor  with  short-circuited  secondary. 

Therefore,  such  motors  with  commutator  always  require 
power-factor  compensation,  by  shifting  the  brushes  or  choosing 
the  impressed  voltage  so  as  to  be  anti-inductive. 

Of  the  voltage  supplied  to  the  secondary  through  commutator 
and  brushes,  a  component  in  phase  with  the  induced  voltage 
lowers  the  speed,  a  component  in  opposition  raises  the  speed, 
and  by  varying  the  commutator  supply  voltage,  speed  control 
of  such  an  induction  motor  can  be  produced  in  the  same  manner 
and  of  the  same  character,  as  produced  in  a  direct-current  motor 


INDUCTION  MOTOR  91 

by  varying  the  field  excitation.  Good  constants  can  be  secured, 
if  in  addition  to  the  energy  component  of  impressed  voltage,  used 
for  speed  control,  a  suitable  anti-inductive  wattless  component 
is  used. 

However,  this  type  of  motor  in  reality  is  not  an  induction 
motor  any  more,  but  a  shunt  motor  or  series  motor,  and  is  more 
fully  discussed  in  Chapter  XIX,  on  "General  Alternating-current 
Motors." 

59.  Suppose,  however,  that  in  addition  to  the  secondary  wind- 
ing connected  to  commutator  and  brushes,  a  short-circuited 
squirrel-cage  winding  is  used  on  the  secondary.  Instead  of 
this,  the  commutator  segments  may  be  shunted  by  resistance, 
which  gives  the  same  effect,  or  merely  a  squirrel-cage  winding 
used,  and  on  one  side  an  end  ring  of  very  high  resistance  em- 
ployed, and  the  brushes  bear  on  this  end  ring,  which  thus  acts 
as  commutator. 

In  either  case,  the  motor  is  an  induction  motor,  and  has  the 
essential  characteristics  of  the  induction  motor,  that  is,  a  slip,  s, 
from  synchronism,  which  increases  with  the  load;  however, 
through  the  commutator  an  exciting  current  can  be  fed  into  the 
motor  from  a  full-frequency  voltage  supply,  and  in  this  case,  the 
current  supplied  over  the  commutator  does  not  meet  the  full- 
frequency  reactance,  xi,  of  the  secondary,  but  only  the  low-fre- 
quency reactance,  sxi,  especially  if  the  commutated  winding  is  in 
the  same  slots  with  the  squirrel-cage  winding :  the  short-circuited 
squirrel-cage  winding  acts  as  a  short-circuited  secondary  to  the 
high-frequency  pulsation  of  the  commutated  current,  and  there- 
fore makes  the  circuit  non-inductive  for  these  high-frequency 
pulsations,  or  practically  so.  That  is,  in  the  short-circuited  con- 
ductors, local  currents  are  induced  equal  and  opposite  to  the 
high-frequency  component  of  the  commutated  current,  and  the 
total  resultant  of  the  currents  in  each  slot  thus  is  only  the  low- 
frequency  current. 

Such  short-circuited  squirrel  cage  in  addition  to  the  commu- 
tated winding,  makes  the  use  of  a  commutator  practicable  for 
power-factor  control  in  the  induction  motor.  It  forbids,  how- 
ever, the  use  of  the  commutator  for  speed  control,  as  due  to  the 
short-circuited  winding,  the  motor  must  run  at  the  slip,  s,  corre- 
sponding to  the  load  as  induction  motor.  The  voltage  impressed 
upon  the  commutator,  and  its  phase  relation,  or  the  brush  posi- 
tion, thus  must  be  chosen  so  as  to  give  only  magnetizing,  but 


92  ELECTRICAL  APPARATUS 

no  speed  changing  effects,  and  this  leaves  only  one  degree  of 
freedom. 

The  foremost  disadvantage  of  this  method  of  secondary  excita- 
tion of  an  induction  motor,  by  a  commutated  winding  in  addi- 
tion to  the  short-circuited  squirrel  cage,  is  that  secondary  excita- 
tion is  advantageous  for  power-factor  control  especially  in 
slow-speed  motors  of  very  many  poles,  and  in  such,  the  commuta- 
tor becomes  very  undesirable,  due  to  the  large  number  of  poles. 
With  such  motors,  it  therefore  is  preferable  to  separate  the 
commutator,  placing  it  on  a  small  commutating  machine  of  a 
few  poles,  and  concatenating  this  with  the  induction  motor.  In 
motors  of  only  a  small  number  of  poles,  in  which  a  commutator 
would  be  less  objectionable,  power-factor  compensation  is  rarely 
needed.  This  is  the  foremost  reason  that  this  type  of  motor 
(the  Heyland  motor)  has  found  no  greater  application. 


CHAPTER  V 
SINGLE-PHASE  INDUCTION  MOTOR 

60.  As  more  fully  discussed  in  the  chapters  on  the  single-phase 
induction  motor,  in  "  Theoretical  Elements  of  Electrical  Engineer- 
ing" and  "  Theory  and  Calculation  of  Alternating-current 
Phenomena,"  the  single-phase  induction  motor  has  inherently, 
no  torque  at  standstill,  that  is,  when  used  without  special  device 
to  produce  such  torque  by  converting  the  motor  into  an  unsym- 
metrical  ployphase  motor,  etc.  The  magnetic  flux  at  standstill 
is  a  single-phase  alternating  flux  of  constant  direction,  and  the 
line  of  polarization  of  the  armature  or  secondary  currents,  that 
is,  the  resultant  m.m.f.  of  the  armature  currents,  coincides  with 
the  axis  of  magnetic  flux  impressed  by  the  primary  circuit. 
When  revolving,  however,  even  at  low  speeds,  torque  appears  in 
the  single-phase  induction  motor,  due  to  the  axis  of  armature 
polarization  being  shifted  against  the  axis  of  primary  impressed 
magnetic  flux,  by  the  rotation.  That  is,  the  armature  currents, 
lagging  behind  the  magnetic  flux  which  induces  them,  reach 
their  maximum  later  than  the  magnetic  flux,  thus  at  a  time  when 
their  conductors  have  already  moved  a  distance  or  an  angle 
away  from  coincidence  with  the  inducing  magnetic  flux.  That  is, 

if  the  armature  currents  lag  ~  =  90°  beyond  the  primary  main 

flux,  and  reach  their  maximum  90°  in  time  behind  the  magnetic 
flux,  at  the  slip,  s,  and  thus  speed  (1  —  s),  they  reach  their  maxi- 
mum in  the  position  (1  —  s)  ~  =  90  (1  —  s)  electrical  degrees 

ft 

behind  the  direction  of  the  main  magnetic  flux.  A  component 
of  the  armature  currents  then  magnetizes  in  the  direction  at 
right  angles  (electrically)  to  the  main  magnetic  flux,  and  the 
armature  currents  thus  produce  a  quadrature  magnetic  flux, 
increasing  from  zero  at  standstill,  to  a  maximum  at  synchronism, 
and  approximately  proportional  to  the  quadrature  component  of 
the  armature  polarization,  P: 

Psin(l  -8)1' 
93 


94  ELECTRICAL  APPARATUS 

The  torque  of  the  single-phase  motor  then  is  produced  by  the 
action  of  the  quadrature  flux  on  the  energy  currents  induced  by 
the  main  flux,  and  thus  is  proportional  to  the  quadrature  flux. 

At  synchronism,  the  quadrature  magnetic  flux  produced  by 
the  armature  currents  becomes  equal  to  the  main  magnetic  flux 
produced  by  the  impressed  single-phase  voltage  (approximately, 
in  reality  it  is  less  by  the  impedance  drop  of  the  exciting  current 
in  the  armature  conductors)  and  the  magnetic  disposition  of  the 
single-phase  induction  motor  thus  becomes  at  synchronism  iden- 
tical with  that  of  the  polyphase  induction  motor,  and  approxi- 
mately so  near  synchronism. 

The  magnetic  field  of  the  single-phase  induction  motor  thus 
may  be  said  to  change  from  a  single-phase  alternating  field  at 
standstill,  over  an  unsymmetrical  rotating  field  at  intermediate 
speeds,  to  a  uniformly  rotating  field  at  full  speed. 

At  synchronism,  the  total  volt-ampere  excitation  of  the  single- 
phase  motor  thus  is  the  same  as  in  the  polyphase  motor  at  the 
same  induced  voltage,  and  decreases  to  half  this  value  at  stand- 
still, where  only  one  of  the  two  quadrature  components  of 
magnetic  flux  exists.  The  primary  impedance  of  the  motor  is 
that  of  the  circuits  used.  The  secondary  impedance  varies 
from  the  joint  impedance  of  all  phases,  at  synchronism,  to  twice 
this  value  at  standstill,  since  at  synchronism  all  the  secondary 
circuits  correspond  to  the  one  primary  circuit,  while  at  stand- 
still only  their  component  parallel  with  the  primary  circuit 
corresponds. 

61.  Hereby  the  single-phase  motor  constants  are  derived  from 
the  constants  of  the  same  motor  structure  as  polyphase  motor. 

Let,  in  a  polyphase  motor: 

Y  =  g   —  jb    =  primary  exciting  admittance; 

ZQ  =  TO  +  JXQ  =  primary  self-inductive  im- 
pedance; 

Zi  =  n  +  jxi  =  secondary  self-inductive  im- 
pedance (reduced  to  the  pri- 
mary by  the  ratio  of  turns, 
in  the  usual  manner); 

the  characteristic  constant  of  the  motor  then  is: 

0  =  Y  (ZQ  +  ZO.  (1) 

The  total,  or  resultant  admittance  respectively  impedance  of 


SINGLE-PHASE  INDUCTION  MOTOR 


95 


the  motor,  that  is,  the  joint  admittance  respectively  impedance  of 
all  the  phases,  then  is: 
In  a  three-phase  motor: 

F°  =  3  F,     1 

Zo°  =   l/3  Z0,  1  (2) 


In  a  quarter-phase  motor: 

F°  =  2  F, 

Zo°  =  ^  Z0, 


(3) 


In  the  same  motor,  as  single-phase  motor,  it  is  then:  at  syn- 
chronism:    s  =  0: 

F'  =  F°, 


Z'o  =  2  Z0°, 


hence  the  characteristic  constant : 

q/  v'  C7f      i     y/ 

V  0  =    I     \L  §  -\-  L  \ 

at  standstill :     s  =  1: 


Z'        _    O    'Z   0 
o  —  "  £*  o  ) 

^f/       070 

hence,  the  characteristic  constant: 


(4) 


(5) 


.  (6) 


(7) 


approximately,  that  is,  assuming  linear  variation  of  the  constants 
with  the  speed  or  slip,  it  is  then:  at  slip,  s: 


Z  o   =    2  ZQ, 


This  gives,  in  a  three-phase  motor: 
Y'~3Y(l- 


(8) 


(9) 


96  ELECTRICAL  APPARATUS 

In  a  quarter-phase  motor: 

,  F'=2F(1- 

Z'Q  =   ZQ 


(10) 


Thus  the  characteristic  constant,  $',  of  the  single-phase  motor 
is  higher,  that  is,  the  motor  inferior  in  its  performance  than  the 
polyphase  motor;  but  the  quarter-phase  motor  makes  just  as 
good — or  poor — a  single-phase  motor  as  the  three-phase  motor. 

62.  The  calculation  of  the  performance  curves  of  the  single- 
phase  motor  from  its  constants,  then,  is  the  same  as  that  of  the 
polyphase  motor,  except  that: 

In  the  expression  of  torque  and  of  power,  the  term  (1  —  s) 
is  added,  which  results  from  the  decreasing  quadrature  flux,  and 
it  thus  is: 

Torque : 

r  =  T  (i  -  s) 

=  (1  -  s)  a,e\  (11) 

Power : 

P'  =  P  (I  -  s) 

=  (1  -  s)2«ie2.  (12) 

However,  these  expressions  are  approximate  only,  as  they 
assume  a  variation  of  the  quadrature  flux  proportional  to  the 
speed. 

63.  As  the  single-phase  induction  motor  is  not  inherently 
self -starting,  starting  devices  are  required.     Such  are: 

(a)  Mechanical  starting. 

As  in  starting  a  single-phase  induction  motor  it  is  not  neces- 
sary, as  in  a  synchronous  motor,  to  bring  it  up  to  full  speed,  but 
the  motor  begins  to  develop  appreciable  torque  already  at  low 
speed,  it  is  quite  feasible  to  start  small  induction  motors  by  hand, 
by  a  pull  on  the  belt,  etc.,  especially  at  light-load  and  if  of  high- 
resistance  armature. 

(6)  By  converting  the  motor  in  starting  into  a  shunt  or  series 
motor. 

This  has  the  great  objection  of  requiring  a  commutator,  and  a 
commutating-machine  rotor  winding  instead  of  the  common 
induction-motor  squirrel-cage  winding.  Also,  as  series  motor, 
the  liability  exists  in  the  starting  connection,  of  running  away; 


SINGLE-PHASE  INDUCTION  MOTOR  97 

as  shunt  motor,  sparking  is  still  more  severe.     Thus  this  method 
is  used  to  a  limited  extent  only. 

(c)  By  shifting  the  axis  of  armature  or  secondary  polarization 
against  the  axis  of  inducing  magnetism. 

This  requires  a  secondary  system,  which  is  electrically  un- 
symmetrical  with  regards  to  the  primary  system,  and  thus,  since 
the  secondary  is  movable  with  regards  to  the  primary,  requires 
means  of  changing  the  secondary  circuit,  that  is,  commutator 
brushes  short-circuiting  secondary  coils  in  the  position  of  effective 
torque,  and  open-circuiting  them  in  the  position  of  opposing  torque. 

Thus  this  method  leads  to  the  various  forms  of  repulsion 
motors,  of  series  and  of  shunt  characteristic. 

It  has  the  serious  objection  of  requiring  a  commutator  and  a 
corresponding  armature  winding;  though  the  limitation  is  not 
quite  as  great  as  with  the  series  or  shunt  motor,  since  in  the  re- 
pulsion motors  the  armature  current  is  an  induced  secondary 
current,  and  the  armature  thus  independent  of  the  primary 
system  regards  current,  voltage  and  number  of  turns. 

(d)  By  shifting  the  axis  of  magnetism,  that  is  producing  a 
magnetic  flux  displaced  in  phase  and  in  position  from  that  in- 
ducing the  armature  currents,  in  other  words,   a  quadrature 
magnetic  flux,  such  as  at  speed  is  being  produced  by  the  rotation. 

This  method  does  not  impose  any  limitation  on  stator  and 
rotor  design,  requires  no  commutator  and  thus  is  the  method 
almost  universally  employed. 

It  thus  may  be  considered  somewhat  more  in  detail. 

The  infinite  variety  of  arrangements  proposed  for  producing 
a  quadrature  or  starting  flux  can  be  grouped  into  three  classes: 

A.  Phase-splitting  Devices. — The  primary  system  of  the  single- 
phase  induction  motor  is  composed  of  two  or  more  circuits 
displaced   from   each   other  in   position   around  the  armature 
circumference,  and  combined  with  impedances  of  different  in- 
ductance factors  so  as  to  produce  a  phase  displacement  between 
them. 

The  motor  circuits  may  be  connected  in  series,  and  shunted 
by  the  impedance,  or  they  may  be  connected  in  shunt  with  each 
other,  but  in  series  with  their  respective  impedance,  or  they 
may  be  connected  with  each  other  by  transformation,  etc. 

B.  Inductive  Devices. — The  motor  is  excited  by  two  or  more 
circuits  which  are  in  inductive  relation  with  each  other  so  as  to 
produce  a  phase  displacement. 

7 


98  ELECTRICAL  APPARATUS 

This  inductive  relation  may  be  established  outside  of  the  motor 
by  an  external  phase-splitting  device,  or  may  take  place  in  the 
motor  proper. 

C.  Monocydic  Devices. — An   essentially  reactive   quadrature 
voltage  is  produced  outside  of  the  motor,  and  used  to  energize 
a  cross-magnetic  circuit  in  the  motor,  either  directly  through  a 
separate  motor  coil,  or  after  combination  with  the  main  voltage 
to  a  system  of  voltages  of  approximate  three-phase  or  quarter- 
phase  relation. 

D.  Phase  Converter. — By  a  separate  external  phase  converter — 
usually  of  the  induction-machine  type — the  single-phase  supply 
is  converted  into  a  polyphase  system. 

Such  phase  converter  niay  be  connected  in  shunt  to  the  motor, 
or  may  be  connected  in  series  thereto.  -% 

This  arrangement  requires  an  auxiliary  machine,  running  idle, 

however.     It  therefore  is  less  convenient,  but  has  the  advantage 

of  being  capable  of  giving  full  polyphase  torque  and  output  to 

the  motor,  and  thus  would  be  specially  suitable  for  railroading. 

64.  If: 

3>o  =  main    magnetic    flux    of    single-phase 

motor,  that  is,  magnetic  flux  produced 

by  the  impressed  single-phase  voltage, 

and 

$  =  auxiliary    magnetic    flux    produced    by 

starting  device,  and  if 
w  =  space  angle  between  the  two  fluxes,  in 

electrical  degrees,  and 
$  =  time  angle  between  the  two  fluxes, 

then  the  torque  of  the  motor  is  proportional  to : 

T  =  a<£<i>o  sin  co  sin  <£;  (13) 

in  the  same  motor  as  polyphase  motor,  with  the  magnetic  flux, 
3>o,  the  torque  is : 

To  =  a$02;  (14) 

thus  the  torque  ratio  of  the  starting  device  is : 

T        $ 

t  =  T  =  <r sin  w  sin  ^'  ^ 

or,  if: 

$'  =  quadrature  flux  produced  by  the  starting  device,  that  is, 


SINGLE-PHASE  INDUCTION  MOTOR  99 

component  of  the  auxiliary  flux,  in  quadrature  to  the  main  flux, 
<l>o,  in  time  and  in  space,  it  is  : 
Single-phase  motor  starting  torque  : 

T  =  a$'$0,  (16) 

and  starting-torque  ratio: 


As  the  magnetic  fluxes  are  proportional  to  the  impressed  vol- 
tages, in  coils  having  the  same  number  of  turns,  it  is:  starting 
torque  of  single-phase  induction  motor: 


T  —  be0e  sin  co  sin  <f> 
=  be0e', 


and,  starting-torque  ratio: 

t  =  —  sin  03  sin  0 

e'  (19) 

where : 

e0  =  impressed  single-phase  voltage, 
e    =  voltage  impressed  upon  the  auxiliary  or 
starting  winding,  reduced  to  the  same 
number  of  turns  as  the  main  winding, 
and 
e'  =  quadrature  component,  in  time  and  in 

space,  of  this  voltage,  e, 

and  the  comparison  is  made  with  the  torque  of  a  quarter-phase 
motor  of  impressed  voltage,  e0,  and  the  same  number  of  turns. 

Or,  if  by  phase-splitting,  monocyclic  device,  etc.,  two  voltages, 
ei  and  e2,  are  impressed  upon  the  two  windings  of  a  single-phase 
induction  motor,  it  is: 
Starting  torque: 

T  =  betfz  sin  co  sin  0  (20) 

and,  starting-torque  ratio: 

t  =  4jf  sin  o>  sin  0,  (21) 

where  eQ  is  the  voltage  impressed  upon  a  quarter-phase  motor, 
with  which  the  single-phase  motor  torque  is  compared,  and  all 


100  ELECTRICAL  APPARATUS 

these  voltages,  ej,  62,  eo,  are  reduced  to  the  same  number  of  turns 
of  the  circuits,  as  customary. 
If  then: 

Q  =  volt-amperes  input  of  the  single-phase 

motor  with  starting  device,  and 
Qo  =  volt-amperes  input  of  the  same  motor 
with  polyphase  supply, 

2  =  |  (22) 

Qo 

is  the  volt-ampere  ratio,  and  thus: 

H  .  v  =  l  ^         (23) 

is  the  ratio  of  the  apparent  starting-torque  efficiency  of  the 
single-phase  motor  with  starting  device,  to  that  of  the  same 
motor  as  polyphase  motor,  v  may  thus  be  called  the  apparent 
torque  efficiency  of  the  single-phase  motor-starting  device. 

In  the  same  manner  the  apparent  power  efficiency  of  the  start- 
ing device  would  result  by  using  the  power  input  instead  of  the 
volt-ampere  input. 

65.  With  a  starting  device  producing  a  quadrature  voltage,  e', 

t  =  -  (24) 

e0 

is  the  ratio  of  the  quadrature  voltage  to  the  main  voltage,  and 
also  is  the  starting-torque  ratio. 
The  quadrature  flux: 

ef  =  too  (25) 

requires  an  exciting  current,  equal  to  t  times  that  of  the  main 
voltage  in  the  motor  without  starting  device,  the  exciting  current 
at  standstill  is: 

,  v>      e°y° 
0      =  ~2~ 

and  in  the  motor  with  starting  device  giving  voltage  ratio,  t, 
the  total  exciting  current  at  standstill  thus  is: 


SINGLE-PHASE  INDUCTION  MOTOti  101 

and  thus,  the  exciting  admittance : 

F'--yCl.+  '0;  (27) 

in  the  same  manner,  the  secondary  impedance  at  standstill  is : 

*>-TTi         -  '       ™ 

and  thus: 

in  the  single-phase  induction  motor  with  starting  device  pro- 
ducing at  standstill  the  ratio  of  quadrature  voltage  to  main 
voltage : 

t  =  e- 

the  constants  are,  at  slip,  s: 

.      .         -        y^yoji.lttju 

Z'o=  2Z0°, 


(29) 


However,  these  expressions  (29)  are  approximate  only,  as  they 
assume  linear  variation  with  s,  and  furthermore,  they  apply  only 
under  the  condition,  that  the  effect  of  the  starting  device  does 
not  vary  with  the  speed  of  the  motor,  that  is,  that  the  voltage 
ratio,  t,  does  not  depend  on  the  effective  impedance  of  the  motor. 
This  is  the  case  only  with  a  few  starting  devices,  while  many 
depend  upon  the  effective  impedance  of  the  motor,  and  thus 
with  the  great  change  of  the  effective  impedance  of  the  motor 
with  increasing  speed,  the  conditions  entirely  change,  so  that  no 
general  equations  can  be  given  for  the  motor  constants. 

66.  Equations  (18)  to  (23)  permit  a  simple  calculation  of  the 
starting  torque,  torque  ratio  and  torque  efficiency  of  the  single- 
phase  induction  motor  with  starting  device,  by  comparison  with 
the  same  motor  as  polyphase  motor,  by  means  of  the  calculation 
of  the  voltages,  e'f  e\,  e2,  etc.,  and  this  calculation  is  simply  that 
of  a  compound  alternating-current  circuit,  containing  the  induc- 
tion motor  as  an  effective  impedance.  That  is,  since  the  only 
determining  factor  in  the  starting  torque  is  the  voltage  impressed 
upon  the  motor,  the  internal  reactions  of  the  motor  do  not  come 
into  consideration,  but  the  motor  merely  acts  as  an  effective 
impedance.  Or  in  other  words,  the  consideration  of  the  internal 


102 


ELECTRICAL  APPARATUS 


reaction  of  the  motor  is  eliminated  by  the  comparison  with  the 
polyphase  motor. 

In  calculating  the  effective  impedance  of  the  motor  at  stand- 
still, we  consider  the  same  as  an  alternating-current  transformer, 
and  use  the  equivalent  circuit  of  the  transformer,  as  discussed 
in  Chapter  XVII  of  "  Theory  and  Calculation  of  Alternating- 
current  Phenomena."  That  is,  the  induction  motor  is  con- 
sidered as  two  impedances,  Z0  and  Zi,  connected  in  series  to  the 


z, 


FIG.  35. — The  equivalent  circuit  of  the  induction  motor. 

impressed  voltage,  with  a  shunt  of  the  admittance,  Y0,  between 
the  two  impedances,  as  shown  in  Fig.  35. 
The  effective  impedance  then  is: 


Z  =  Z0  +  - 


1 


(30) 


approximately,  this  is : 


(31) 


This  approximation  (31),  is  very  close,  if  Zi  is  highly  inductive, 
as  a  short-circuited  low-resistance  squirrel  cage,  but  ceases  to  be 
a  satisfactory  approximation  if  the  secondary  is  of  high  resistance, 
for  instance,  contains  a  starting  rheostat. 

As  instances  are  given  in  the  following  the  correct  values  of  the 
effective  impedance,  Z,  from  equation  (30),  the  approximate 
value  (31),  and  their  difference,  for  a  three-phase  motor  without 
starting  resistance,  with  a  small  resistance,  with  the  resistance 
giving  maximum  torque  at  standstill,  and  a  high  resistance: 


YO: 


SINGLE-PHASE  INDUCTION  MOTOR 


z,: 


103 


0.01  -  0.1 3  0.1  +  0.3  j  0.1     +  0.3 3  0.195  +  0.592  j  0.2     +  0.6  j  -  0.005  -  0.008 3 

0.25  +  0.3  3  0.336  +  0.506 /  0.35  +  0.6  /  -  0.014  -  0.004  j 

0.6    +  0.3j  0.661  +0.620/  0.7    +  0.6j  -  0.039  +  0.020 ;' 

1.6     +0.3 3  1.552+ 0.804  /  1.7     +  0.6 j  -  0.148  +  0.204  j 

A.  PHASE-SPLITTING  DEVICES 
Parallel  Connection 

67.  Let  the  motor  contain  two  primary  circuits  at  right  angles 
(electrically)  in  space  with  each  other,  and  of  equal  effective 
impedance: 

Z  =  r  +  jx. 
These  two  motor  circuits  are  connected  in  parallel  with  each 


FIG.  36. — Diagram  of  phase-splitting  device  with  parallel  connection  of 

motor  circuits. 

other  between  the  same  single-phase  mains  of  voltage,  eo,  but 
the  first  motor  circuit  contains  in  series  the  impedance 

Zi  =  TI  +  jxi, 
the  second  motor  circuit  the  impedance : 

Z*  =  rz  +  jxt, 

as  shown  diagrammatically  in  Fig.  36. 
The  two  motor  currents  then  are: 


T  Q  A 

Jl  =      ~    and 


(32) 
(33) 


the  two  voltages  across  the  two  motor  coils: 


104  ELECTRICAL  APPARATUS 

E1  =  I&    and     #2  =  72Z 

" 


/O/l\ 


and  the  phase  angle  between  EI  and  E%  is  given  by  : 

7     I  •  7 

m  (cos  ^  +  j  sin  0)  =  „       „-•  (35) 

Z  +  Z2 

Denoting  the  absolute  values  of  the  voltages  and  currents  by 
small  letters,  it  is: 

T  =  betfz  sin  <£;  (36) 


in  the  motor  as  quarter-phase  motor,  with  voltage,  eQ,  impressed 
per  circuit,  it  is: 

To  =  foo2,  (37) 

hence,  the  torque  ratio: 

•         6162     «       .  /oo\ 

t  =  — •£  sin  9.  1><J°/ 

The  current  per  circuit,  in  the  machine  as  quarter-phase  motor, 
is: 

to  =  ~>  (39) 

hence  the  volt-amperes: 

Qo  =  2  Colo,  (40) 

while  the  volt-amperes  of  the  single-phase  motor,  inclusive  start- 
ing impedances,  are : 

Q  =  e»i,  (41) 

thus: 

(42) 


and,  the  apparent  torque  efficiency  of  the  starting  device: 


t 

v  =  - 


q         eQiz 
68.  As  an  instance,  consider  the  motor  of  effective  impedance  : 

Z  =r+jx  =  0.1  +  0.3.7, 
thus: 

z  =  0.316, 


SINGLE-PHASE  INDUCTION  MOTOR  105 

and  assume,  as  the  simplest  case,  a  resistance,  a  =  0.3,  inserted 
in  series  to  the  one  motor  circuit.     That  is: 

Zl  =  0,  (44) 

Z2  =  a. 
It  is  then: 


/OQN.T 

~ 


r+jx       0.1  +  0.3  j      '         r  +  a+jx       0.4  +  0.3  j 
=  e0(l  -  3j),  =  «o(1.6  -  1.2  j); 

(33):  /  =  e0(2.6  -4.2J), 

i  =  4.94  e0; 


(34): 


ei  =  Co,         e2  =  0.632  e0; 
(35)  :       m  (cos  *  +  j  sin  *)  =  - 


=  0.52  +  0.36  j, 
0.36 


sin  0  =  0.57; 

(38):  *  =  0.36; 

(43)  :  v  =  0.46. 

Thus  this  arrangement  gives  46  per  cent.,  or  nearly  half  as 
much  starting  torque  per  volt-ampere  taken  from  the  supply 
circuit,  as  the  motor  would  give  as  polyphase  motor. 

However,  as  polyphase  motor  with  low-resistance  secondary, 
the  starting  torque  per  volt-ampere  input  is  low. 

With  a  high-resistance  motor  armature,  which  on  polyphase 
supply  gives  a  good  apparent  starting-torque  efficiency,  v  would 
be  much  lower,  due  to  the  lower  angle,  0.  In  this  case,  however, 
a  reactance,  -\-jat  would  give  fairly  good  starting-torque  efficiency. 

In  the  same  manner  the  effect  of  reactance  or  capacity  inserted 
into  one  of  the  two  motor  coils  can  be  calculated. 

As  instances  are  given,  in  Fig.  37,  the  apparent  torque  efficiency, 
v,  of  the  single-phase  induction-motor  starting  device  consisting 
of  the  insertion,  in  one  of  the  two  parallel  motor  circuits,  of 
various  amounts  of  reactance,  inductive  or  positive,  and  capacity 


106  ELECTRICAL  APPARATUS 

or  negative,  for  a  low  secondary  resistance  motor  of  impedance: 

Z  =  0.1  +  0.3  j 
and  a  high  resistance  armature,  of  the  motor  impedance : 

Z  =  0.3  +  0.1  j 
resistance  inserted  into  the  one  motor  circuit,  has  the  same  effect 


1.0 

.8 

.6 

.4 

Z  = 

3+lj 

(Z 

=  1+3 

I 

a= 

+.2 

X 

-1-8  4 

7    4- 

6     + 

5     4 

4     4 

3     4 

2     4- 

1 

^ 

Z=l+3j 

(Z 

=  3+lJ 

1     + 

2     + 

3     +J4     + 

5    + 

6     +J7  +8 

^ 

0 
+.2 

Cfi 

PAC 

TY 

/ 

/ 

+.4 

IN 

DUG' 

'ANC 

E  (F 

ESIS 

FAN 

:E> 

j/ 

4-.6 

^^ 

/I 

+.8 

"^ 

•«^Z^i 

3  +  1; 

^ 

'    1 

4-1.0 

1 

4-1.2 

\ 

/ 

4-1.4 

\ 

/ 

+1.6 

\ 

1 

+1.8 

\ 

<.' 

+  sj 

...  —  1 

2 

+2.0 

FIG.  37. — Apparent  starting-torque  efficiencies  of  phase-splitting  device, 
parallel  connection  of  motor  circuits. 

in  the  first  motor,  as  positive  reactance  in  the  second  motor,  and 
inversely. 

69.  Higher  values  of  starting-torque  efficiency  are  secured  by 
the  use  of  capacity  in  the  one,  and  inductance  in  the  other  motor 
circuit.  It  is  obvious  that  by  resistance  and  inductance  alone, 
90°  phase  displacement  between  the  two  component  currents, 
and  thus  true  quarter-phase  relation,  can  not  be  reached. 

As  resistance  consumes  energy,  the  use  of  resistance  is  justified 


SINGLE-PHASE  INDUCTION  MOTOR  107 

only  due  to  its  simplicity  and  cheapness,  where  moderate  start- 
ing torques  are  sufficient,  and  thus  the  starting-torque  efficiency 
less  important.  For  producing  high  starting  torque  with  high 
starting-torque  efficiency,  thus,  only  capacity  and  inductance 
would  come  into  consideration. 

Assume,  then,  that  the  one  impedance  is  a  capacity: 

x2  =  —  k}     or:     Z2  =  —  jk,  (45) 

while  the  other,  x\,  may  be  an  inductance  or  also  a  capacity,  what- 
ever may  be  desired  : 

Zi  =  +jxlt  (46) 

where  x\  is  negative  for  a  capacity. 
It  is,  then: 

(35)  :  m  (cos  <£  +  j  sin  0)  = 

r+j  (xi  +  a?)  _  [r2  -  (xi  +  x)(k  -  x)]  +  jrxjc     ,.„. 
r-j(k-x)   '  r2  +  (k-x)z 

True  quadrature  relation  of  the  voltages,  e\  and  62,  or  angle, 

7T 

0  =  s;  requires: 

Zi 

cos  0  =  0, 
thus: 

(xi  +  a;)  (/c  -  x)  =  r2  (48) 

and  the  two  voltages,  e\  and  e*,  are  equal,  that  is,  a  true  quarter- 
phase  system  of  voltages  is  produced,  if  in 

(34):  [Z  +  ZJ  =  [Z  +  Z2], 

where  the  [     ]  denote  the  absolute  values. 
This  gives: 

r2  +  (Xl  +  xY  =  r*  +  (k  -  x)\ 
or: 

xi  +  x  =  k  -  x,  (49) 

hence,  by  (48): 

Xi  -\-  x  =  k  —  x  =  r, 
k  =  r  +  x, 
Xi  =  r  —  x. 


(go) 


Thus,  if  x  >  r,  or  in  a  low-resistance  motor,  the  second  reactance, 
j  also  must  be  a  capacity. 


108  ELECTRICAL  APPARATUS 

70.  Thus,  let: 
in  a  low-resistance  motor : 

Z  =  r  +  jx  =  0.1  +  0.3.?, 
k  =  0.4,  xi  =  -  0.2, 

Z2  =  -0.4J,      Zl  =  -0.2.7, 

that  is,  both  reactances  are  capacities. 

(34) :  ei  =  e2  =  2.23  e0, 

*  =  5, 

that  is,  the  torque  is  five  times  as  great  as  on  true  quarter-phase 


supply, 


0.1  +  0.1  j'  '2    ''0.1  -Q.I  j} 

/  =  10  e0  =  i, 


that  is,  non-inductive,  or  unity  power-factor. 

f0  =  |  =  3.166o, 
L 

q  =  1.58, 
0  =  3.16, 

that  is,  the  apparent  starting-torque  efficiency,  or  starting  torque 
per  volt-ampere  input,  of  the  single-phase  induction  motor  with 
starting  devices  consisting  of  two  capacities  giving  a  true  quarter- 
phase  system,  is  3.16  as  high  as  that  of  the  same  motor  on  a 
quarter-phase  voltage  supply,  and  the  circuit  is  non-inductive 
in  starting,  while  on  quarter-phase  supply,  it  has  the  power- 
factor  31.6  per  cent,  in  starting. 
In  a  high-resistance  motor: 

Z  =  0.3  +  0.1  j, 
it  is: 

k  =  0.4,  xi  =  0.2, 

Z2  =  -  0.4  j,    Z2  =  +0.2J, 

that  is,  the  one  reactance  is  a  capacity,  the  other  an  inductance. 

e\  =  ez  =  0.743  BQ, 
t    =  0.555, 
i    =  3.33  e0, 

•  O     "I  £* 

^o  =  o.lo  60, 
q    =  0.527, 
v    =  1.055, 


SINGLE-PHASE  INDUCTION  MOTOR 


109 


that  is,  the  starting-torque  efficiency  is  a  little  higher  than  with 
quarter-phase  supply.  In  other  words: 

This  high-resistance  motor  gives  5.5  per  cent,  more  torque 
per  volt-ampere  input,  with  unity  power-factor,  on  single-phase 
supply,  than  it  gives  on  quarter-phase  supply  with  95  per  cent, 
power-factor. 

The  value  found  for  the  low-resistance  motor,  t  =  5,  is  how- 
ever not  feasible,  as  it  gives:  e\  =  e%  =  2.23  eo,  and  in  a  quarter- 
phase  motor  designed  for  impressed  voltage,  e0,  the  impressed 
voltage,  2.23  eQ,  would  be  far  above  saturation.  Thus  the  motor 
would  have  to  be  operated  at  lower  supply  voltage  single-phase, 
and  then  give  lower  t,  though  the  same  value  of  v  =  3.16.  At 
ei  =  e2  =  eo,  the  impressed  voltage  of  the  single-phase  circuit 
would  be  about  45  per  cent,  of  eQ,  and  then  it  would  be:  t  =  1. 

Thus,  in  the  low-resistance  motor,  it  would  be  preferable  to 
operate  the  two  motor  circuits  in  series,  but  shunted  by  the  two 
different  capacities  producing  true  quarter-phase  relation. 

Series  Connection 

71.  The  calculation  of  the  single-phase  starting  of  a  motor 
with  two  coils  in  quadrature  position,  shunted  by  two  impedances 


FIG.  38. — Diagram  of  phase-splitting  device  with  series  connection  of  motor 

circuits 

of  different  power-factor,  as  shown  diagrammatically  in  Fig.  38, 
can  be  carried  out  in  the  same  way  as  that  of  parallel  connection, 
except  that  it  is  more  convenient  in  series  connection  to  use  the 
term  " admittance"  instead  of  impedance. 

That  is,  let  the  effective  admittance  per  motor  coil  equal : 


110  ELECTRICAL  APPARATUS 

and  the  two  motor  coils  be  shunted  respectively  by  the  admit- 
tances: 


2  =  02  - 
it  is  then: 


F  +  Fi       Y  +  F2 
the  current  consumed  by  the  motor,  and  : 


(52) 
(53) 

,rT!7  (54) 

the  voltages  across  the  two  motor  circuits. 

The  phase  difference  between  EI  and  Ez  thus  is  given  by 

Y  +  Y 

m  (cos  0  +  j  sin  </>)  =  y-.  -y- >  (55) 

and  herefrom  follows  ty  q  and  0. 

As  instance  consider  a  motor  of  effective  admittance  per  cir- 
cuit: 

7  =  g  -  jb  =  1  -  3j, 

with  the  two  circuits  connected  in  series  between  single-phase 
mains  of  voltage,  eo,  and  one  circuit  shunted  by  a  non-inductive 
resistance  of  conductance,  grj. 

What  value  of  gi  gives  maximum  starting  torque,  and  what 
is  this  torque? 

It  is: 

^  (56) 


^  +  flfi  -  jb       g  -  jb 

/r  ,N  .       ET  e0-(flf  —  j&)  ^       eo  (g  +  gi  -  jb)  .        , 

(54)"      ^1  =  ^  =  -' 


hence : 

tan  6  = 


Q  (Q 


sin  0  =  —r=         -     *  (58) 

2  +  [g  (g  +  g,)  +  62]2 


SINGLE-PHASE  INDUCTION  MOTOR  111 

and  thus: 

etfz  sin  <f)  gib 

t   —    » =~ 


— -,  (59) 

y  -r  Q\r  +  462 

and  for  maximum,  t: 

Wi  ~     ' 
thus: 

0i  =  2  Vg2  +  b2 

=  2  y  =  6.32,  (60) 

or,  substituting  back: 

(59):  j  =  —A^  =  0.18.  (61) 

4  (</  +  y) 

As  in  single-phase  operation,  the  voltage,  e0,  is  impressed  upon 
the  two  quadrature  coils  in  series,  each  coil  receives  only  about 

— 4=.     Comparing  then  the  single-phase  starting  torque  with  that 

v  2 

of  a  quarter-phase  motor  of  impressed  voltage,  —/='  it  is : 

t  =  0.36. 

The  reader  is  advised  to  study  the  possibilities  of  capacity 
and  reactance  (inductive  or  capacity)  shunting  the  two  motor 
coils,  the  values  giving  maximum  torque,  those  giving  true 
quarter-phase  relation,  and  the  torque  and  apparent  torque 
efficiencies  secured  thereby. 

B.  INDUCTIVE  DEVICES 
External  Inductive  Devices 

72.  Inductively  divided  circuit :  in  its  simplest  form,  as  shown 
diagrammatically  in  Fig.  39,  the  motor  contains  two  circuits 
at  right  angles,  of  the  same  admittance. 

The  one  circuit  (1)  is  in  series  with  the  one,  the  other  (2)  with 
the  other  of  two  coils  wound  on  the  same  magnetic  circuit,  M. 
By  proportioning  the  number  of  turns,  n\  and  n2,  of  the  two  coils, 
which  thus  are  interlinked  inductively  with  each  other  on  the 
external  magnetic  circuit,  M,  a  considerable  phase  displacement 


112 


ELECTRICAL  APPARATUS 


between  the  motor  coils,  and  thus  starting  torque  can  be  pro- 
duced, especially  with  a  high-resistance  armature,  that  is,  a 
motor  witl^  starting  rheostat. 

A  full  discussion  and  calculation  of  this  device  is  contained  in 
the  paper  on  the  " Single-phase  Induction  Motor,"  page  63, 
A.  I.  E.  E.  Transactions,  1898. 


FIG.    39. — External   inductive 
device. 


FIG.  40. — Diagram  of  shading  coil. 


Internal  Inductive  Devices 

The  exciting  system  of  the  motor  consists  of  a  stationary  pri- 
mary coil  and  a  stationary  secondary  coil,  short-circuited  upon 
itself  (or  closed  through  an  impedance),  both  acting  upon  the 
revolving  secondary. 

The  stationary  secondary  can  either  cover  a  part  of  the  pole 
face  excited  by  the  primary  coil,  and  is  then  called  a  "shading 
coil,"  or  it  has  the  same  pitch  as  the  primary,  but  is  angularly 
displaced  therefrom  in  space,  by  less  than  90°  (usually  45°  or  60°), 
and  then  has  been  called  accelerating  coil. 

The  shading  coil,  as  shown  diagrammatically  in  Fig.  40,  is 
the  simplest  of  all  the  single-phase  induction  motor-starting 
devices,  and  therefore  very  extensively  used,  though  it  gives 
only  a  small  starting  torque,  and  that  at  a  low  apparent  starting- 
torque  efficiency.  It  is  almost  exclusively  used  in  very  small 
motors  which  require  little  starting  torque,  such  as  fan  motors, 
and  thus  industrially  constitutes  the  most  important  single- 
phase  induction  motor-starting  device. 

73.  Let,  all  the  quantities  being  reduced  to  the  primary  num- 
ber of  turns  and  frequency,  as  customary  in  induction  machines : 

Z0  =  rQ  +  JXQ  =  primary  self-inductive  impedance, 
Y   =  g   —  jb    =  primary  exciting  admittance  of  unshaded  poles 
(assuming  total  pole  unshaded), 


SINGLE-PHASE  INDUCTION  MOTOR  113 

Y'  =  g'  —  jb'  =  primary  exciting  admittance  of  shaded  poles 
(assuming  total  pole  shaded). 

If  the  reluctivity  of  the  shaded  portion  of  the  pole  is  the  same 
as  that  of  the  unshaded,  then  F'  =  F;  in  general,  if 

b  =  ratio  of  reluctivity  of  shaded  to  unshaded  portion  of 
pole, 

F'  =  bY, 

b  either  =  1,  or,  sometimes,  b  >  1,  if  the  air  gap  under  the 
shaded  portion  of  the  pole  is  made  larger  than  that  under  the 
unshaded  portion. 

YI  =  gi  —  jbi  =  self-inductive  admittance  of  the  revolving 
secondary  or  armature, 

YZ  =  gz  —  jbz  =  self-inductive  admittance  of  the  stationary 
secondary  or  shading  coil,  inclusive  its  exter- 
nal circuit,  where  such  exists. 

Z0,  FI  and  F2  thus  refer  to  the  self-inductive  impedances,  in 
which  the  energy  component  is  due  to  effective  resistance,  and 
F  and  Y'  refer  to  the  mutual  inductive  impedances,  in  which  the 
energy  component  is  due  to  hysteresis  and  eddy  currents. 

a  =  shaded  portion  of  pole,  as  fraction  of  total  pole;  thus 

(1  —  a)  =  unshaded  portion  of  pole. 
If: 

eQ  =  impressed  single-phase  voltage, 

EI  =  voltage  induced  by  flux  in  unshaded  portion  of  pole, 
EZ  =  voltage  induced  by  flux  in  shaded  portion  of  pole, 
/o  =  primary  current, 
it  is  then: 

€Q  =  EI  -f-  EZ  H~  ZO/D-  (62) 

The  secondary  current  in  the  armature  under  the  unshaded 
portion  of  the  pole  is : 

1 1  =  EiY,.  (63) 

The  primary  exciting  current  of  the  unshaded  portion  of  the 
pole: 

/oo  =  ^j-~,  (64) 

thus: 

/o  =  /i  +  /oo  =  #,  {  F!  +  1  -— -} .  (65) 

I  i  —  a  i 


114 


ELECTRICAL  APPARATUS 


The  secondary  current  under  the  shaded  portion  of  the  pole  is : 
I'i  =  #2^1.  (66) 

The  current  in  the  shading  coil  is: 

h  =  E2Y2.  (67) 

The  primary  exciting  current  of  the  shaded  portion  of  the  pole 


00 


E2bY 


a 


thus: 


/O    =    /'l   +  /OO  +  /2    == 


'i+rr+r.   J 


from  (65)  and  (69)  follows: 


Y,  +  -  Y  +  Y, 


# 


=  m  (cos  <j>  +  j  sin  </>), 


(68) 
(69) 

(70) 


1  -  a 

and  this  gives  the  angle,  <£,  of  phase  displacement  between  the  two 
component  voltages,  $1  and  E%. 

If,  as  usual,  b  =  1,  and 

if  a  =  0.5,  that  is,  half  the  pole  is  shaded,  it  is: 

,.:      l-Hr^-1-    •     -  (7i) 

74.  Assuming  now,  as  first  approximation,  Z0  =  0,  that  is, 
neglecting  the  impedance  drop  in  the  single-phase  primary  coil — 
which  obviously  has  no  influence  on  the  phase  difference  between 
the  component  voltages,  and  the  ratio  of  their  values,  that  is, 
on  the  approximation  of  the  devices  to  polyphase  relation — then 
it  is: 

Ti1     _1_    7?  n    •  ff7c>\ 

JC/i   "I"   jC/2    —    &Q)  \'£) 

thus,  from  (70) : 


(73) 


TF     —   f> 

y,  +  b-  Y  +  Y, 

TJT 

Y   i  Y(b  i     l    \  \  y*" 

Y  (a   '    1  -  a) 

V      1 

yi    '    1  -a 

v    ,   v/b           1     \        v  ' 

YL  '    r\a    '    1-  J    '    y2 

SINGLE-PHASE  INDUCTION  MOTOR 


115 


or,  for: 


b  =  1;  a  =  0.5; 

+  2  Y  +  F 


2  Fi  +  4  F  +  IV 


(74) 


2  F!  +  4  F  +  F2' 

and  the  primary  current,  or  single-phase  supply  current  is,  by 
substituting  (73)  into  (65): 

,    b  ,r  ,    , 


or,  for: 


6  =  1;  a  =  0.5: 
(F,  +  2  F)(F1  +  2  7  + 


(75) 


(76) 


and  herefrom  follows,  by  reducing  to  absolute  values,  the  torque, 
torque  ratio,  volt-ampere  input,  apparent  torque  efficiency,  etc. 
Or,  denoting: 


1  -a 
Y  +  \  Y  +  F2  =  F', 


(77) 


it  is: 

ET              y/ 

(70)  :                    •—  =  y-  =  m  (cos  0  +  j  sin  0)  ;                      (78) 

*J  ~  yo   i    y/' 
(73):                               Y  e^y* 

(79) 
csm 

E*  ~  yo  +  y^, 

^7.^^  •                            T  n  —    -                 i 

sn 


and  for  a  quarter-phase  motor,  with  voltage  — -p=  impressed  per 


116  ELECTRICAL  APPARATUS 

circuit,  neglecting  the  primary  impedance,  z0)  to  be  comparable 
with  the  shaded-coil  single-phase  motor,  it  is: 


=  «?/Y  +  Y,/, 

T  A  6°2 

T"  =  A      ' 


thus: 


f 

v  =  --• 

q 

75.  As  instances  are  given  in  the  following  table  the  compo- 
nent voltages,  e\  and  e2,  the  phase  angle,  $,  between  them,  the 
primary  current,  io,  the  torque  ratio,  t,  and  the  apparent  starting- 
torque  efficiency,  v,  for  the  shaded-pole  motor  with  the  constants: 

Impressed  voltage:  eQ    =  100; 

Primary  exciting  admittance:  Y    =  0.001  —  0.01  j. 

6  =  1,  that  is,  uniform  air  gap. 

a  =  0.5,  that  is,  half  the  pole  is  shaded. 

And  for  the  three  motor  armatures  : 

Low  resistance:  FI  =  0.01  —  0.03  j, 
Medium  resistance:  FI  =  0.02  -  0.02  j, 
High  resistance:  FI  =  0.03  -  0.01  j; 

and  for  the  three  kinds  of  shading  coils  : 

Low  resistance:  F2  =  0.01  —  0.03  j, 

Medium  resistance:  F2  =  0.02  -  0.02  j, 
High  resistance:  F2  =  0.03  -  0.01  j. 

As  seen  from  this  table,  the  phase  angle,  <£,  and  thus  the  start- 
ing torque,  t,  are  greatest  with  the  combination  of  low-resistance 
armature  and  high-resistance  shading  coil,  and  of  high-resistance 
armature  with  low-resistance  shading  coil;  but  in  the  first  case 
the  torque  is  in  opposite  direction  —  accelerating  coil  —  from  what 


SINGLE-PHASE  INDUCTION  MOTOR  117 

it  is  in  the  second  case — lagging  coil.  In  either  case,  the  torque 
efficiency  is  low,  that  is,  the  device  is  not  suitable  to  produce 
high  starting-torque  efficiencies,  but  its  foremost  advantage  is 
the  extreme  simplicity. 

The  voltage  due  to  the  shaded  portion  of  the  pole,  e2,  is  less 
than  that  due  to  the  unshaded  portion,  e\,  and  thus  a  somewhat 
higher  torque  may  be  produced  by  shading  more  than  half  of 
the  pole:  a  >  0.5. 

A  larger  air  gap:  6  >  1,  under  the  shaded  portion  of  the  pole, 
or  an  external  non-inductive  resistance  inserted  into  the  shad- 
ing coil,  under  certain  conditions  increases  the  torque  somewhat — 
at  a  sacrifice  of  power-factor — particularly  with  high-resistance 
armature  and  low-resistance  shading  coil. 

e0  =  100  volts;  a  =  0.5;  b  =  1;  Y  =  0.001  -  0.01  j. 
YI'.        Y2:        cii        ez:          0:          t0:  t,:  v: 


X    10-2     X    1Q-2 

per  cent. 

per  cent, 

1 

-3/1 

-  3.7 

38.3 

61.8 

+  1 

.9 

1.97 

+  1 

.56 

+  4.07 

2 

-  2,7 

40.3 

60.2 

+  11 

.0 

2.07 

+  9 

.28 

+23 

.00 

3 

-M 

42.0 

59.8 

+21 

.5 

2.17 

+  18 

.36 

+43 

.70 

2 

-•3/1 

_3y 

37.2 

62.9 

-  4 

.a 

1.70 

-  3.52 

-  9 

.65 

2 

-2/ 

38.5 

61.7 

+  6 

.2 

1.76 

+  5, 

.12 

+  13 

.60 

3 

-  i/ 

39.2 

62.0 

+  17 

.3 

1.80 

+  14. 

44 

+37 

.40 

3 

-Mi 

-3.7 

37.6 

63.0 

-11 

.9 

1.66 

-  9 

76 

-25 

.80 

2 

-2.7 

37.8 

62.5 

-  0 

.8 

1.66 

-  0 

.66 

-   1 

.75 

3 

-  Ij 

37.4 

63.0 

+  10 

.3 

1.64 

+  8. 

44 

+22 

.60 

Monocyclic  Starting  Device 

76.  The  monocyclic  starting  device  consists  in  producing  ex- 
ternally to  the  motor  a  system  of  polyphase  voltages  with  single- 
phase  flow  of  energy,  and  impressing  it  upon  the  motor,  which  is 
wound  as  polyphase  motor. 

If  across  the  single-phase  mains  of  voltage,  e,  two  impedances 
of  different  inductance  factors,  Zi  and  Z2,  are  connected  in  series, 
as  shown  diagrammatically  in  Fig.  41,  the  two  voltages,  EI  and 
$2,  across  these  two  impedances  are  displaced  in  phase  from  each 
other,  thus  forming  with  the  main  voltage  a  voltage  triangle. 
The  altitude  of  this  triangle,  or  the  voltage,  E0,  between  the  com- 


118 


ELECTRICAL  APPARATUS 


mon  connection  of  the  two  impedances,  and  a  point  inside  of  the 
main  voltage,  e  (its  middle,  if  the  two  impedances  are  equal),  is 
a  voltage  in  quadrature  with  the  main  voltage,  and  is  a  teazer 
voltage  or  quadrature  voltage  of  the  monocyclic 
system,  e,  EI,  Ez,  that  is,  it  is  of  limited  energy 
and  drops  if  power  is  taken  off  from  it.  (See 
Chapter  XIV.) 

Let  then,  in  a  three-phase  wound  motor,  oper- 
ated single-phase  with  monocyclic  starting  device, 
and  shown  diagrammatically  in  Fig.  42 : 

e  =  voltage    impressed     between    single-phase 
FIG.  41.       lines, 
Monocyclic          /  =  current  in  single-phase  lines, 

Y  =  effective  admittance  per  motor  circuit, 

YI,  EI  and  7'i,  and  Y2,  E2  and  7'2  =  admittance,  voltage  and 
current  respectively,  in  the  two  impedances  of  the  mono- 
cyclic  starting  device, 
I 


FIG.  42. — Three-phase  motor  with  monocyclic  starting  device. 

/i,  /2  and  /3  =  currents  in  the  three  motor  circuits. 

EQ  and  70  =  voltage  and  current  of  the  quadrature  circuit  from 

the  common  connection  of  the  two  impedances, 

to  the  motor. 


SINGLE-PHASE  INDUCTION  MOTOR 


119 


It  is  then,  counting  the  voltages  and  currents  in  the  direction 
indicated  by  the  arrows  of  Fig.  42: 

I*  =  /'.  -  /'•  =  h-h;  (81) 

substituting: 

f  i  =  ViYlt 


gives : 
thus: 


I'z  — 


Tfi    T7"       /  Tf 


(82) 


r  =  v^~  r^  =  m  (cos  0  +  j  sin 

2  Jl  -T   J 


(83) 


This  gives  the  phase  angle,  0,  between  the  voltages,  EI  and  Ez, 
of  the  monocyclic  triangle.     Since: 


it  is,  by  (83) : 


=  e 


Ei  +  E2  =  e, 
F2+  F 

,         (84) 
(85) 

F!  +  F2  +  2  F' 
F,  +  F 

y     i    y    _i_  o  F' 

and  the  quadrature  voltage: 


r,  -  r, 


2  Fi  +  F2  +  2  F 


(86) 


and  the  total  current  input  into  the  motor,  inclusive  starting 
device : 

i  =  rl  +  il  +  /3 

f(F1+F)(F2+F)  | 

1     F!  +  F2  +  2  F 
FXF2  +  2  F(F,  +  F2)  +  3  F2 
Fl  +  y2  +  2  F 

As  with  the  balanced  three-phase  motor,  the  quadrature  com- 
ponent of  voltage  numerically  is  «  \/3>  it  is,  when  denoting  by : 


120  ELECTRICAL  APPARATUS 

E03'  the  numerical  value  of  the  imaginary  term  of  EQ-,  the  torque 
ratio  is: 

t  =  ^'  (88) 

e\/3 

The  volt-ampere  ratio  is: 

''   '''  " 


thus  the  apparent  starting-torque  efficiency: 

v  =  t,  (90) 

etc. 

77.  Three  cases  have  become  of  special  importance  : 

(a)  The  resistance-reactance  monocyclic  starting  device  ;  where 
one  of  the  two  impedances,  Zi  and  Z2,  is  a  resistance,  the  other  an 
inductance.  This  is  the  simplest  and  cheapest  arrangement, 
gives  good  starting  torque,  though  a  fairly  high  current  consump- 
tion and  therefore  low  starting-torque  efficiency,  and  is  therefore 
very  extensively  used  for  starting  single-phase  induction  motors. 
After  starting,  the  monocyclic  device  is  cut  out  and  the  power 
consumption  due  to  the  resistance,  and  depreciation  of  the  power- 
factor  due  to  the  inductance,  thereby  avoided. 

This  device  is  discussed  on  page  333  of  "  Theoretical  Elements 
of  Electrical  Engineering"  and  page  253  of  ''Theory  and  Calcu- 
lation of  Alternating-current  Phenomena." 

(6)  The  "  condenser  in  the  tertiary  circuit,"  which  may  be 
considered  as  a  monocyclic  starting  device,  in  which  one  of  the 
two  impedances  is  a  capacity,  the  other  one  is  infinity.  The 
capacity  usually  is  made  so  as  to  approximately  balance  the  mag- 
netizing current  of  the  motor,  is  left  in  circuit  after  starting,  as 
it  does  not  interfere  with  the  operation,  does  not  consume  power, 
and  compensates  for  the  lagging  current  of  the  motor,  so  that 
the  motor  has  practically  unity  power-factor  for  all  loads.  This 
motor  gives  a  moderate  starting  torque,  but  with  very  good  start- 
ing-torque efficiency,  and  therefore  is  the  most  satisfactory  single- 
'phase  induction  motor,  where  very  high  starting  torque  is  not 
needed.  It  was  extensively  used  some  years  ago,  but  went  out 
of  use  due  to  the  trouble  with  the  condensers  of  these  early  days, 
and  it  is  therefore  again  coming  into  use,  with  the  development 
of  the  last  years,  of  a  satisfactory  condenser. 


SINGLE-PHASE  INDUCTION  MOTOR  121 

The  condenser  motor  is  discussed  on  page  249  of  "  Theory  and 
Calculation  of  Alternating-current  Phenomena." 

(c)  The  condenser-inductance  monocyclic  starting  device. 
By  suitable  values  of  capacity  and  inductance,  a  balanced  three- 
phase  triangle  can  be  produced,  and  thereby  a  starting  torque 
equal  to  that  of  the  motor  on  three-phase  voltage  supply,  with 
an  apparent  starting-torque  efficiency  superior  to  that  of  the 
three-phase  motor. 

Assuming  thus: 

YI  =  -\-jbi  =  capacity, 
Y2  =  —jb2  =  inductance, 
Y   =  g-jb. 

If  the  voltage  triangle,  e,  EI,  Ez,  is  a  balanced  three-phase  tri- 
angle, it  is : 

e  , 


(91) 


(92) 

<H 

Substituting  (91)  and  (92)  into  (83),  and  expanding  gives: 
(62  -  h  +  2  b)  V3  -  j  (bz  +  b,  -  2  g  x/3)  =  0; 

thus: 

b2  -  &!  +  26  =  0, 
bz  +  &i  —  20  \/3  =  0; 
hence : 

b.^Vs  +  ft,  j 

02  =  9  V3  —  b]  I 
thus,  if: 

b>  g  \/3, 

the  second  reactance,  Z2,  must  be  a  capacity  also;  if 

b  <  g  \/3, 

only  the  first  reactance,  Zi,  is  a  capacity,  but  the  second  is  an 
inductance. 

78.  Considering,  as  an  instance,  a  low-resistance  motor,  and  a 
high-resistance  motor: 

(a)  (b) 

Y  =  g  -  jb  =  1  -  3  j,  Y  =  g  -  jb  =  3  -  j, 


122  ELECTRICAL  APPARATUS 

it  is: 

61  =  4.732,  capacity,  61  =  6.196,  capacity, 

62  =  —1.268,  capacity,  bz  =  4.196,  inductance. 
It  is,  by  (86)  and  (92) 


thus: 

t  =  1,  as  was  to  be  expected. 
/s  =  e  (g  -  jb), 
i*  =  e  \/02  +  b2 
=  3.16  e; 

it  is,  however,  by  (87)  : 

/  =  e(3g  -jb)', 
thus: 

i  =  4.243  e,  i  =  9.06  e, 

and  by  (89)  : 

q  =  0.448,  q  =  0.956, 

thus: 

v  =  2.232,  v  =  1.046. 

Further  discussion  of  the  various  single-phase  induction  motor- 
starting  devices,  and  also  a  discussion  of  the  acceleration  of  the 
motor  with  the  starting  device,  and  the  interference  or  non-inter- 
ference of  the  starting  device  with  the  quadrature  flux  and  thus 
torque  produced  in  the  motor  by  the  rotation  of  the  armature,  is 
given  in  a  paper  on  the  "  Single-phase  Induction  Motor/'  A.  I. 
E.  E.  Transactions,  1898,  page  35,  and  a  supplementary  paper  on 
"  Notes  on  Single-phase  Induction  Motors,"  A.  I.  E.  E.  Trans- 
actions, 1900,  page  25. 


CHAPTER  VI 

INDUCTION-MOTOR  REGULATION  AND  STABILITY 

1.  VOLTAGE  REGULATION  AND  OUTPUT 

79.  Load  and  speed  curves  of  induction  motors  are  usually 
calculated  and  plotted  for  constant-supply  voltage  at  the  motor 
terminals.  In  practice,  however,  this  condition  usually  is  only 
approximately  fulfilled,  and  due  to  the  drop  of  voltage  in  the 
step-down  transformers  feeding  the  motor,  in  the  secondary  and 
the  primary  supply  lines,  etc.,  the  voltage  at  the  motor  terminals 
drops  more  or  less  with  increase  of  load.  Thus,  if  the  voltage 
at  the  primary  terminals  of  the  motor  transformer  is  constant, 
and  such  as  to  give  the  rated  motor  voltage  at  full-load,  at  no- 
load  the  voltage  at  the  motor  terminals  is  higher,  but  at  overload 
lower  by  the  voltage  drop  in  the  internal  impedance  of  the  trans- 
formers. If  the  voltage  is  kept  constant  in  the  center  of  distri- 
bution, the  drop  of  voltage  in  the  line  adds  itself  to  the  imped- 
ance drop  in  the  transformers,  and  the  motor  supply  voltage 
thus  varies  still  more  between  no-load  and  overload. 

With  a  drop  of  voltage  in  the  supply  circuit  between  the  point 
of  constant  potential  and  the  motor  terminals,  assuming  the  cir- 
cuit such  as  to  give  the  rated  motor  voltage  at  full-load,  the 
voltage  at  no-load  and  thus  the  exciting  current  is  higher,  the 
voltage  at  overload  and  thus  the  maximum  output  and  maximum 
torque  of  the  motor,  and  also  the  motor  impedance  current,  that 
is,  current  consumed  by  the  motor  at  standstill,  and  thereby  the 
starting  torque  of  the  motor,  are  lower  than  on  a  constant-poten- 
tial supply.  Hereby  then  the  margin  of  overload  capacity  of  the 
motor  is  reduced,  and  the  characteristic  constant  of  the  motor, 
or  the  ratio  of  exciting  current  to  short-circuit  current,  is  in- 
creased, that  is,  the  motor  characteristic  made  inferior  to  that 
given  at  constant  voltage  supply,  the  more  so  the  .higher  the 
voltage  drop  in  the  supply  circuit. 

Assuming  then  a  three-phase  motor  having  the  following  con- 
stants: primary  exciting  admittance,  Y  =  0.01  —  0.1  j',  primary 
self-inductive  impedance,  Z0  =  0.1  +  0.3  j;  secondary  self-induc- 

123 


124  ELECTRICAL  APPARATUS 

tive  impedance,  Zi  =  0.1  +  0.3  j;  supply  voltage,  e0  =  110  volts, 
and  rated  output,  5000  watts  per  phase. 
Assuming  this  motor  to  be  operated: 

1.  By  transformers  of  about  2  per  cent,  resistance  and  4  per 
cent,  reactance  voltage,  that  is,  transformers  of  good  regulation, 
with  constant  voltage  at  the  transformer  terminals. 

2.  By  transformers  of  about  2  per  cent,  resistance  and  15  per 
cent,  reactance  voltage,  that  is,  very  poorly  regulating  trans- 
formers, at  constant  supply  voltage  at  the  transformer  primaries. 

3.  With   constant  voltage   at  the   generator  terminals,   and 
about  8  per  cent,  resistance,  40  per  cent,  reactance  voltage  in 
line  and  transformers  between  generator  and  motor. 

This  gives,  in  complex  quantities,  the  impedance  between  the 
motor  terminals  and  the  constant  voltage  supply: 

1.  Z  =  0.04  +  0.08  j, 

2.  Z  =  0.04  +  0.3  j, 

3.  Z  =  0.16  +  0.8  j. 

It  is  assumed  that  the  constant  supply  voltage  is  such  as  to 
give  110  volts  at  the  motor  terminals  at  full-load. 

The  load  and  speed  curves  of  the  motor,  when  operating  under 
these  conditions,  that  is,  with  the  impedance,  Z,  in  series  between 
the  motor  terminals  and  the  constant  voltage  supply,  e\,  then  can 
be  calculated  from  the  motor  characteristics  at  constant  termi- 
nal voltage,  BQ,  as  follows: 

At  slip,  s,  and  constant  terminal  voltage,  e0,  the  current  in  the 
motor  is  i0,  its  power-factor  p  =  cos  0.  The  effective  or  equiva- 
lent impedance  of  the  motor  at  this  slip  then  is  z°  =  g-,  and,  in 

complex  quantities,  Z°  =  ~  (cos  6  +  j  sin  0),  and  the  total  im- 

IQ 

pedance,  including  that  of  transformers  and  line,  thus  is : 
Z,  =  Z°  +  Z  =  (~  cos  e  +  r]  +  j(^  sin  0  +  x] , 

Uo  Mo 

or,  in  absolute  values: 


and,  at  the  supply  voltage,  ei,  the  current  thus  is 


INDUCTION-MOTOR  REGULATION 


125 


and  the  voltage  at  the  motor  terminals  is: 


eo 


If  e0  is  the  voltage  required  at  the  motor  terminals  at  full-load, 
and  ^o°  the  current,  21°  the  total  impedance  at  full-load,  it  is: 


y  o  _        . 
10    -' 


"\ 

sE 

\\ 

• 

7000 

6000 

Kftfln 

Y  =  0.01-  0.1  j      Z=  0.1  +  0.3  j 
TRANSFORMER  IMPEDANCE    Z0=  0.04-1-  0.08  j 
CONSTANT  PRIMARY  POTENTIAL  114.1  VOLTS 

/ 

/ 

/ 

/ 

-.  AMP.VOLT8 
o  PERCENT 

E. 

YI.F. 

T  MO 

FOR  i 

ERMlf 

AL8 

A 

V 

•••• 

i 

-       - 

—  -^— 

"SPEE 

—             __ 
) 

•• 

. 
/ 

/ 

^ 

100 

—    I— 

1  

^    / 

% 

4000 
3000 
2000 
1000 

/ 

/ 

^ 

^ 

80 

/ 

/ 

70 

>, 

/ 

X 

60 

** 

7 

-6 

X 

50 

/ 

S 

x 

40 

/ 

.s* 

X 

30 

/ 

^ 

^ 

SO 

.. 

1* 

^ 

00 

_LO_ 
0 

/ 

1( 

DO 

2( 

DO 

3( 

P    WER 

00     I     4C 

OUTPJUT 

K)      1     50 

K) 

6C 

FIG.  43. — Induction-motor  load  curves  corresponding  to  110  volts  at  motor 
terminals  at  5000  watts  load. 

hence,  the  required  constant  supply  voltage  is: 


and  the  speed  and  torque  curves  of  the  motor  under  this  condi- 
tion then  are  derived  from  those  at  constant  supply  voltage,  eo, 


e'o 


by  multiplying  all  voltages  and  currents  by  the  factor  —  •->  that 

Co 

is,  by  the  ratio  of  the  actual  terminal  voltage  to  the  full-load 
terminal  voltage,  and  the  torque  and  power  by  multiplying  with 


126 


ELECTRICAL  APPARATUS 


the  square  of  this  ratio,  while  the  power-factors  and  the  efficien- 
cies obviously  remain  unchanged. 

In  this  manner,  in  the  three  cases  assumed  in  the  preceding, 
the  load  curves  are  calculated,  and  are  plotted  in  Figs.  43,  44, 
and  45. 

80.  It  is  seen  that,  even  with  transformers  of  good  regulation, 
Fig.  43,  the  maximum  torque  and  the  maximum  power  are  ap- 


Y  =  0.01  -0.1  JO    Z  =  0.1  +  0.3  j 
TRANSFORMER  IMPEDANCE,  Z0  =  0.04  +0.3  j 
CONSTANT   PRIMARY  POTENTIAL    121  VOLTS 

ORQUE 
N.  WATTS 

-J 

'SOOOj 

". 

7000 

"z 

§| 

/ 

2 

6000. 

-120. 

no 

—       • 

—        __ 

E.M 

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—  ' 

[_MOT 

ORjj 

RMIN, 

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—  ^, 

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—             — 

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_LOQ 
90 

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—            — 

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/ 

/ 

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5Q 

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£> 

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10 



/ 

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/ 

1 

00 

20 

30 

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WO 

:R  OUTPUT 
4000 

50 

30 

GO 

)0 

0 

0 

FIG.  44. — Induction-motor  load  curves  corresponding  to  110  volts  at  motor 
terminals  at  5000  watts  load. 

preciably  reduced.  The  values  corresponding  to  constant  termi- 
nal voltage  are  shown,  for  the  part  of  the  curves  near  maximum 
torque  and  maximum  power,  in  Figs.  43,  44,  and  45. 

In  Figs.  46,  47,  48,  and  49  are  given  the  speed-torque  curves 
of  the  motor,  for  constant  terminal  voltage,  Z  =  0,  and  the 
three  cases  above  discussed;  in  Fig.  46  for  short-circuited 
secondaries,  or  running  condition;  in  Fig.  47  for  0.15  ohm;  in 
Fig.  48  for  0.5  ohm;  and  in  Fig.  49  for  1.5  ohms  additional  re- 
sistance inserted  in  the  armature.  As  seen,  the  line  and  trans- 
former impedance  very  appreciably  lowers  the  torque,  and 


INDUCTION-MOTOR  REGULATION 


127 


in 

8000 
7000 
6000 

Y=  O.C1  -  0.1  j       Z  =  0.1  -1-  0.3  j 
CIRCUIT    IMPEDANCE,    Z0=.16+.8J 
CONSTANT  GENERATOR  POTENTIAL  144.5  VOLTS 

'~\ 

i 

-«  AMP.VOLT8 
p  PERCENT 

E.M. 

.  AT 

MOTC 

)R  T£ 

/ 

•^^ 

^v. 

*»s^ 

/ 

x, 

110 

5000 
4000 
3000 
2000 

1000 
0 

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SP 

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•  

_100- 
90 

/ 

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V 

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/ 

60 

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/ 

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x 

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10 

/ 

1 

00' 

2C 

00 

3( 

POW 

00 

R    OUTPU1 

4000 

50 

)0 

60 

K) 

0 

FIG.  45. — Induction-motor  load  curves  corresponding  to  110  volts  at  motor 
terminals  at  500  watts  load. 


FIG.  46. — Induction-motor  speed  torque  characteristics  with  short-circuited 

secondary. 


128 


ELECTRICAL  APPARATUS 


especially  the  starting  torque,  which,  with  short-circuited  arma- 
ture, in  the  case  3  drops  to  about  one-third  the  value  given  at 
constant  supply  voltage. 


FIG.  47. — Induction-motor  speed  torque  characteristics  with  a  resistance  of 
0.15  ohm  in  secondary  circuit. 


Zo-o 

V 

RQUE 
SYN., 
ATTS 

8000 
7300 
CCOO 
5GCO 
4000 
3000 
20CO 
1000 
0 

ZQ  ~~  ' 

•04  +  0.0s7^ 

•x^ 

. 

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J^ 

004* 

D.33 

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^ 



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\\ 

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1.0     0 

9        0 

8          0 

SLIP 

7       0 

;RACTIO 
.6           0 

iOFSY 
.5        0 

CHRONI 

.4  0 

IM 
.3         0 

2         0 

^ 

1      0' 

FIG.  48. — Induction-motor  speed  torque  characteristics  with  a  resistance  of 
0.5  ohm  in  secondary  circuit. 

It  is  interesting  to  note  that  in  Fig.  48,  with  a  secondary 
resistance  giving  maximum  torque  in  starting,  at  constant  ter- 


INDUCTION-MOTOR  REGULATION 


129 


minal  voltage,  with  high  impedance  in  the  supply,  the  starting 
torque  drops  so  much  that  the  maximum  torque  is  shifted  to 
about  half  synchronism. 

In  induction  motors,  especially  at  overloads  and  in  starting, 
it  therefore  is  important  to  have  as  low  impedance  as  pos- 
sible between  the  point  of  constant  voltage  and  the  motor 
terminals. 


1.0      0.1 


0.8        0.7        0.6       0.5       0.4        0.3 
Slip  Fraction  of  Synchronism 


0.2 


FIG.  49.  —  Induction-motor  speed  current  characteristics  with  a  resistance  of 
1.5  ohms  in  secondary  circuit. 


In  Table  I  the  numerical  values  of  maximum  power,  maxi- 
mum torque,  starting  torque,  exciting  current  and  starting 
current  are  given  for  above  motor,  at  constant  terminal  voltage 
and  for  the  three  values  of  impedance  in  the  supply  lines,  for 
such  supply  voltage  as  to  give  the  rated  motor  voltage  of  110 
volts  at  full  load  and  for  110  volts  supply,  voltage.  In  the  first 
case,  maximum  power  and  torque  drop  down  to  their  full-load 
values  with  the  highest  line  impedance,  and  far  below  full-load 
values  in  the  latter  case. 

9 


130 


ELECTRICAL  APPARATUS 


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INDUCTION-MOTOR  REGULATION 


131 


2.  FREQUENCY  PULSATION 

81.  If  the  frequency  of  the  voltage  supply  pulsates  with 
sufficient  rapidity  that  the  motor  speed  can  not  appreciably 
follow  the  pulsations  of  frequency,  the  motor  current  and  torque 
also  pulsate;  that  is,  if  the  frequency  pulsates  by  the  fraction, 
p,  above  and  below  the  normal,  at  the  average  slip,  s,  the  actual 
slip  pulsates  between  s  +  p  and  s  —  p,  and  motor  current  and 


AMP. 

150 
140 
130 
120 
110 
100 
90 
80 
70 
60 

50 
40 
30 
20 

10 
0 

TOKC 
SYN.W 

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8000 

7000 
6000 
5000 
4000 
3000 
2000 
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-2000 
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FREQUENCY  FLUCTUATING  BY  s=±0.02 
OR  5  PER  CENT 

\ 

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or* 

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13  OJ12  OJ11  ollO  0.09  0 

08  0. 

07  0.06  0 

05  0. 

31  0 

03  0.02  0. 

11    0 

FIG.  50. — Effect  of  Frequency  Pulsation  on  Induction  Motor. 

torque  pulsate  between  the  values  corresponding  to  the  slips, 
s  +  p  and  s  —  p.  If  then  the  average  slip  s  <  p,  at  minimum 
frequency,  the  actual  slip,  s  —  p,  becomes  negative;  that  is,  the 
motor  momentarily  generates  and  returns  energy. 

As  instance  are  shown,  in  Fig.  50,  the  values  of  current  and 
of  torque  for  maximum  and  minimum  frequency,  and  for  the 
average  frequency,  for  p  =  0.025,  that  is,  2.5  per  cent,  pulsa- 
tion of  frequency  from  the  average.  As  seen,  the  pulsation  of 
current  is  moderate  until  synchronism  is  approached,  but  be- 


132  ELECTRICAL  APPARATUS 

comes  very  large  near  synchronism,  and  from  slip,  s  =  0.025,  up 
to  synchronism  the  average  current  remains  practically  con- 
stant, thus  at  synchronism  is  very  much  higher  than  the  current 
at  constant  frequency.  The  average  torque  also  drops  some- 
what below  the  torque  corresponding  to  constant  frequency, 
as  shown  in  the  upper  part  of  Fig.  50. 

3.  LOAD  AND  STABILITY 

82.  At  constant  voltage  and  constant  frequency  the  torque 
of  the  polyphase  induction  motor  is  a  maximum  at  some  definite 
speed  and  decreases  with  increase  of  speed  over  that  correspond- 
ing to  the  maximum  torque,  to  zero  at  synchronism;  it  also  de- 
creases with  decrease  of  speed  from  that  at  the  maximum  torque 
point,  to  a  minimum  at  standstill,  the  starting  torque.  This 
maximum  torque  point  shifts  toward  lower  speed  with  increase 
of  the  resistance  in  the  secondary  circuit,  and  the  starting  torque 
thereby  increases.  Without  additional  resistance  inserted  in 
the  secondary  circuit  the  maximum  torque  point,  however,  lies 
at  fairly  high  speed  not  very  far  below  synchronism,  10  to  20 
per  cent,  below  synchronism  with  smaller  motors  of  good  effi- 
ciency. Any  value  of  torque  between  the  starting  torque  and 
the  maximum  torque  is  reached  at  two  different  speeds.  Thus 
in  a  three-phase  motor  having  the  following  constants :  impressed 
e.m.f.,  eQ  =  110  volts;  exciting  admittance,  Y  =  0.01  —  0.1  j; 
primary  impedance,  ZQ  =  0.1  +  0.3  j,  and  secondary  impedance, 
Z1  =  0.1  +  0.3j,  the  torque  of  5.5  synchronous  kw.  is  reached 
at  54  per  cent,  of  synchronism  and  also  at  the  speed  of  94  per 
cent,  of  synchronism,  as  seen  in  Fig.  51. 

When  connected  to  a  load  requiring  a  constant  torque,  irre- 
spective of  the  speed,  as  when  pumping  water  against  a  constant 
head  by  reciprocating  pumps,  the  motor  thus  could  carry  the 
load  at  two  different  speeds,  the  two  points  of  intersection  of  the 
horizontal  line,  L,  in  Fig.  51,  which  represents  the  torque  con- 
sumed by  the  load,  and  the  motor-torque  curve,  D.  Of  these 
two  points,  d  and  c,  the  lower  one,  d,  represents  unstable  con- 
ditions of  operation;  that  is,  the  motor  can  not  operate  at  this 
speed,  but  either  stops  or  runs  up  to  the  higher  speed  point,  c, 
at  which  stability  is  reached.  At  the  lower  speed,  d,  a  momen- 
tary decrease  of  speed,  as  by  a  small  pulsation  of  voltage,  load, 
etc.,  decreases  the  motor  torque,  D,  below  the  torque,  L,  required 
by  the  load,  thus  causes  the  motor  to  slow  down,  but  in  doing 


INDUCTION-MOTOR  REGULATION 


133 


so  its  torque  further  decreases,  and  it  slows  down  still  more, 
loses  more  torque,  etc.,  until  it  comes  to  a  standstill.  Inversely, 
a  momentary  increase  of  speed  increases  the  motor  torque,  D, 
beyond  the  torque,  L,  consumed  by  the  load,  and  thereby  causes 
an  acceleration,  that  is,  an  increase  of  speed.  This  increase  of 
speed,  however,  increases  the  motor  torque  and  thereby  the 
speed  still  further,  and  so  on,  and  the  motor  increases  in  speed 
up  to  the  point,  c,  where  the  motor  torque,  D,  again  becomes 


FIG.  51. — Speed -torque  characteristics  of  induction  motor  and  load  for 
determination  of  the  stability  point. 


equal  to  the  torque  consumed  by  the  load.  A  momentary  in- 
crease of  speed  beyond  c  decreases  the  motor  torque,  D,  and  thus 
limits  itself,  and  inversely  a  momentary  decrease  of  speed  below 
c  increases  the  motor  torque,  D,  beyond  L,  thus  accelerates  and 
recovers  the  speed;  that  is,  at  c  the  motor  speed  is  stable. 

With  a  load  requiring  constant  torque  the  induction  motor 
thus  is  unstable  at  speeds  below  that  of  the  maximum  torque 
point,  but  stable  above  it;  that  is,  the  motor  curve  consists  of 
two  branches,  an  unstable  branch,  from  standstill,  t,  to  the  maxi- 


134  ELECTRICAL  APPARATUS 

mum  torque  point,  m,  and  a  stable  branch,  from  the  maximum 
torque  point,  m,  to  synchronism. 

83.  It  must  be  realized,  however,  that  this  instability  of  the 
lower  branch  of  the  induction-motor  speed  curve  is  a  function  of 
the  nature  of  the  load,  and  as  described  above  applies  only  to  a 
load  requiring  a  constant  torque,  L.  Such  a  load  the  motor 
could  not  start  (except  by  increasing  the  motor  torque  at  low 
speeds  by  resistance  in  the  secondary),  but  when  brought  up  to 
a  speed  above  d  would  carry  the  load  at  speed,  c,  in  Fig.  51.  . 

If,  however,  the  load  on  the  motor  is  such  as  to  require  a 
torque  which  increases  with  the  square  of  the  speed,  as  shown 
by  curve,  C,  in  Fig.  51,  that  is,  consists  of  a  constant  part  p 
(friction  of  bearings,  etc.)  and  a  quadratic  part,  as  when  driving 
a  ship's  propeller  or  driving  a  centrifugal  pump,  then  the  induc- 
tion motor  is  stable  over  the  entire  range  of  speed,  from  standstill 
to  synchronism.  The  motor  then  starts,  with  the  load  repre- 
sented by  curve  C,  and  runs  up  to  speed,  c.  At  a  higher  load, 
represented  by  curve  B,  the  motor  runs  up  to  speed,  6,  and  with 
excessive  overload,  curve  A,  the  motor  would  run  up  to  low 
speed,  point  a,  only,  but  no  overload  of  such  nature  would  stop 
the  motor,  but  merely  reduce  its  speed,  and  inversely,  it  would 
always  start,  but  at  excessive  overloads  run  at  low  speed  only. 
Thus  in  this  case  no  unstable  branch  of  the  motor  curve  exists, 
but  it  is  stable  over  the  entire  range. 

With  a  load  requiring  a  torque  which  increases  proportionally 
to  the  speed,  as  shown  by  C  in  Fig.  52,  that  is,  which  consists 
of  a  constant  part,  p,  and  a  part  proportional  to  the  speed,  as 
when  driving  a  direct-current  generator  at  constant  excitation, 
connected  to  a  constant  resistance  as  load — as  a  lighting  sys- 
tem— the  motor  always  starts,  regardless  of  the  load — provided 
that  the  constant  part  of  the  torque,  p,  is  less  than  the  starting 
torque.  With  moderate  load,  C,  the  motor  runs  up  to  a  speed, 
c,  near  synchronism.  With  very  heavy  load,  A,  the  motor  starts, 
but  runs  up  to  a  low  speed  only.  Especially  interesting  is  the 
case  of  an  intermediary  load  as  represented  by  line  B  in  Fig. 
52.  B  intersects  the  motor-torque  curve,  D,  in  three  points, 
bi,  62,  bs;  that  is,  three  speeds  exist  at  which  the  motor  gives  the 
torque  required  by  the  load:  24  per  cent.,  60  per  cent.,  and  88 
per  cent,  of  synchronism.  The  speeds  fei  and  63  are  stable,  the 
speed  62  unstable.  Thus,  with  this  load  the  motor  starts  from 
standstill,  but  does  not  run  up  to  a  speed  near  synchronism,  but 


INDUCTION-MOTOR  REGULATION 


135 


accelerates  only  to  speed  61,  and  keeps  revolving  at  this  low 
speed  (and  a  correspondingly  very  large  current).  If,  however, 
the  load  is  taken  off  and  the  motor  allowed  to  run  up  to  syn- 
chronism or  near  to  it,  and  the  load  then  put  on,  the  motor  slows 
down  only  to  speed  63,  and  carries  the  load  at  this  high  speed; 
hence,  the  motor  can  revolve  continuously  at  two  different  speeds, 
bi  and  63,  and  either  of  these  speeds  is  stable;  that  is,  a  momen- 
tary increase  of  speed  decreases  the  motor  torque  below  that 


FIG.  52. — Speed  torque  characteristics  of  induction  motor  and  load  for 
determination  of  the  stability  point. 

required  by  the  load,  and  thus  limits  itself,  and  inversely  a  de- 
crease of  motor  speed  increases  its  torque  beyond  that  correspond- 
ing to  the  load,  and  thus  restores  the  speed.  At  the  intermediary 
speed,  62,  the  conditions  are  unstable,  and  a  momentary  increase 
of  speed  causes  the  motor  to  accelerate  up  to  speed  63,  a  momen- 
tary decrease  of  speed  from  bi  causes  the  motor  to  slow  down  to 
speed  61,  where  it  becomes  stable  again.  In  the  speed  range 
between  bz  and  bs  the  motor  thus  accelerates  up  to  63,  in  the 
speed  range  between  62  and  b]  it  slows  down  to  fej. 

For  this  character  of  load,  the  induction-motor  speed  curve, 
D,  thus  has  two  stable  branches,  a  lower  one,  from  standstill,  t, 
to  the  point  n,  and  an  upper  one,  from  point  m  to  synchronism, 


136 


ELECTRICAL  APPARATUS 


where  m  and  n  are  the  points  of  contact  of  the  tangents  from  the 
required  starting  torque,  p,  on  to  the  motor  curve,  D;  these  two 
stable  branches  are  separated  by  the  unstable  branch,  from  n  to 
m,  on  which  the  motor  can  not  operate. 

84.  The  question  of  stability  of  motor  speed  thus  is  a  func- 
tion not  only  of  the  motor-speed  curve  but  also  of  the  character 
of  the  load  in  its  relation  to  the  motor-speed  curve,  and  if  the 
change  of  motor  torque  with  the  change  of  speed  is  less  than  the 
change  of  the  torque  required  by  the  load,  the  condition  is  stable, 

otherwise  it  is  unstable:  that  is.  it  must  be  -j^-  <  -TO  to  give 

«o          ao 

stability,  where  L  is  the  torque  required  by  the  load  at  speed,  S. 


01        02       0,3        0.4 


SP   ED 

05 


06        07        0.8        0.9      l.|o 


FIG.  53. — Speed-torque  characteristic  of  single-phase  induction  motor. 

Occasionally  on  polyphase  induction  motors  on  a  load  as  repre- 
sented in  Fjg.  52  this  phenomenon  is  observed  in  the  form 
that  the  motor  can  start  the  load  but  can  not  bring  it  up  to 
speed.  More  frequently,  however,  it  is  observed  on  single- 
phase  induction  motors  in  which  the  maximum  torque  is  nearer 
to  synchronism,  with  some  forms  of  starting  devices  which  de- 
crease in  their  effect  with  increasing  speed  and  thus  give  motor- 
speed  characteristics  of  forms  similar  to  Fig.  53.  With  a 
torque-speed  curve  as  shown  in  Fig.  53,  even  at  a  load  requiring 
constant  torque,  three  speed  points  may  exist  of  which  the 
middle  one  is  unstable.  In  polyphase. synchronous  motors  and 
converters,  when  starting  by  alternating  current,  that  is,  as 


INDUCTION-MOTOR  REGULATION  137 

induction  machines,  the  phenomenon  is  frequently  observed  that 
the  machine  starts  at  moderate  voltage,  but  does  not  run  up  to 
synchronism,  but  stops  at  an  intermediary  speed,  in  the  neighbor- 
hood of  half  speed,  and  a  considerable  increase  of  voltage,  and 
thereby  of  motor  torque,  is  required  to  bring  the  machine  beyond 
the  dead  point,  or  rather  "dead  range,"  of  speed  and  make  it 
run  up  to  synchronism.  In  this  case,  however,  the  phenomenon 
is  complicated  by  the  effects  due  to  varying  magnetic  reluctance 
(magnetic  locking),  inductor  machine  effect,  etc. 

Instability  of  such  character  as  here  described  occurs  in  elec- 
tric circuits  in  many  instances,  of  which  the  most  typical  is  the 
electric  arc  in  a  constant-potential  supply.  It  occurs  whenever 
the  effect  produced  by  any  cause  increases  the  cause  and  thereby 
becomes  cumulative.  When  dealing  with  energy,  obviously 
the  effect  must  always  be  in  opposition  to  the  cause  (Lenz's 
Law),  as  result  of  the  law  of  conservation  of  energy.  When 
dealing  with  other  phenomena,  however,  as  the  speed-torque 
relation  or  the  volt-ampere  relation,  etc.,  instability  due  to  the 
effect  assisting  the  cause,  intensifying  it,  and  thus  becoming 
cumulative,  may  exist,  and  frequently  does  exist,  and  causes 
either  indefinite  increase  or  decrease,  or  surging  or  hunting,  as 
more  fully  discussed  in  Chapters  X  and  XI,  of  "Theory  and 
Calculation  of  Electric  Circuits." 

4.  GENERATOR  REGULATION  AND  STABILITY 

85.  If  the  voltage  at  the  induction-motor  terminals  decreases 
with  increase  of  load,  the  maximum  torque  and  output  are  de- 
creased the  more  the  greater  the  drop  of  voltage.  But  even  if 
the  voltage  at  the  induction  motor  terminals  is  maintained  con- 
stant, the  maximum  torque  and  power  may  be  reduced  essen- 
tially, in  a  manner  depending  on  the  rapidity  with  which  the 
voltage  regulation  at  changes  of  load  is  effected  by  the  generator 
or  potential  regulator,  which  maintains  constancy  of  voltage,  and 
the  rapidity  with  which  the  motor  speed  can  change,  that  is, 
the  mechanical  momentum  of  the  motor  and  its  load. 

This  instability  of  the  motor,  produced  by  the  generator 
regulation,  may  be  discussed  for  the  case  of  a  load  requiring 
constant  torque  at  all  loads,  though  the  corresponding  pheno- 
menon may  exist  at  all  classes  of  load,  as  discussed  under  3, 
and  may  occur  even  with  a  load  proportional  to  the  square  of 
the  speed,  as  ship  propellers. 


138  ELECTRICAL  APPARATUS 

The  torque  curve  of  the  induction  motor  at  constant  terminal 
voltage  consists  of  two  branches,  a  stable  branch,  from  the 
maximum  torque  point  to  synchronism,  and  an  unstable  branch, 
that  is,  a  branch  at  which  the  motor  can  not  operate  on  a  load 
requiring  constant  torque,  from  standstill  to  maximum  torque. 
With  increasing  slip,  s,  the  current,  i,  in  the  motor  increases.  If 

then  D   =  torque  of  the  motor,  —.—  is  positive  on  the  stable, 

negative  on  the  unstable  branch  of  the  motor  curve,  and  this 
rate  of  change  of  the  torque,  with  change  of  current,  expressed 
as  fraction  of  the  current,  is : 

k      -±^' 

"  D  di  ' 

it  may  be  called  the  stability  coefficient  of  the  motor. 

If  ks  is  positive,  an  increase  of  i,  caused  by  an  increase  of 
slip,  s,  that  is,  by  a  decrease  of  speed,  increases  the  torque,  D,  and 
thereby  checks  the  decrease  of  speed,  and  inversely,  that  is,  the 
motor  is  stable. 

If,  however,  ks  is  negative,  an  increase  of  i  causes  a  decrease 
of  D,  thereby  a  decrease  of  speed,  and  thus  further  increase  of  i 
and  decrease  of  D;  that  is,  the  motor  slows  down  with  increas- 
ing rapidity,  or  inversely,  with  a  decrease  of  i,  accelerates  with 
increasing  rapidity,  that  is,  is  unstable. 

For  the  motor  used  as  illustration  in  the  preceding,  of  the 
constants  e  =  110  volts;  Y  =  0.01  --  0.1  j;  Z0  =  0.1  +  0.3  j, 
Zi  =  0.1  +  0.3  j,  the  stability  curve  is  shown,  together  with 
speed,  current,  and  torque,  in  Fig.  54,  as  function  of  the  output. 
As  seen,  the  stability  coefficient,  ks).  is  very  high  for  light-load, 
decreases  first  rapidly  and  then  slowly,  until  an  output  of  7000 
watts  is  approached,  and  then  rapidly  drops  below  zero;  that  is, 
the  motor  becomes  unstable  and  drops  out  of  step,  and  speed, 
torque,  and  current  change  abruptly,  as  indicated  by  the  arrows 
in  Fig.  54. 

The  stability  coefficient,  k8J  characterizes  the  behavior  of  the 
motor  regarding  its  load-carrying  capacity.  Obviously,  if  the 
terminal  voltage  of  the  motor  is  not  constant,  but  drops  with 
the  load,  as  discussed  in  1,  a  different  stability  coefficient  results; 
which  intersects  the  zero  line  at  a  different  and  lower  torque. 

86.  If  the  induction  motor  is  supplied  with  constant  terminal 
voltage  from  a  generator  of  close  inherent  voltage  regulation 


INDUCTION-MOTOR  REGULATION 


139 


and  of  a  size  very  large  compared  with  the  motor,  over  a  supply 
circuit  of  negligible  impedance,  so  that  a  sudden  change  of 
motor  current  can  not  produce  even  a  momentary  tendency  of 
change  of  the  terminal  voltage  of  the  motor,  the  stability  curve, 
ka,  of  Fig.  54  gives  the  performance  of  the  motor.  If,  however, 


Y  =  0.01-0.lj      Z  =  0.1  +  0.3  j 
CONSTANT  POTENTIAL   HO  VOLTS 
GENERATOR  IMPEDANCE  Z -0.02 +0.5  j 
LINE    Z  =  0.1  +  0.2j 
TRANSFORMER  IMPEDANCE   Z  =  0.04  +  O.lj 

REGULATION  COEFFICIENT  OF  SUPPLY  kr  =  f- 
STABILITY  COEFFICIENT  OF  MOTOR  ks  =  -g-    $ 

STABILITY  COEFFICIENT  OF  SYSTEM  k8-kr~ 
MAXIMUM  OUTPUT  POINT  ® 


\ 


i 


FIG.  54. — Induction-motor  load  curves. 


at  a  change  of  load  and  thus  of  motor  current  the  regulation 
of  the  supply  voltage  to  constancy  at  the  motor  terminals  re- 
quires a  finite  time,  even  if  this  time  is  very  short,  the  maximum 
output  of  the  motor  is  reduced  thereby,  the  more  so  the  more 
rapidly  the  motor  speed  can  change. 

Assuming  the  voltage  control  at  the  motor  terminals  effected 


140  ELECTRICAL  APPARATUS 

by  hand  regulation  of  the  generator  or  the  potential  regulator 
in  the  circuit  supplying  the  motor,  or  by  any  other  method  which 
is  slower  than  the  rate  at  which  the  motor  speed  can  adjust  itself 
to  a  change  of  load,  then,  even  if  the  supply  voltage  at  the 
motor  terminals  is  kept  constant,  for  a  momentary  fluctuation 
of  motor  speed  and  current,  the  supply  voltage  momentarily 
varies,  and  with  regard  to  its  stability  the  motor  corresponds 
not  to  the  condition  of  constant  supply  voltage  but  to  a  supply 
voltage  which  varies  with  the  current,  hence  the  limit  of  stability 
is  reached  at  a  lower  value  of  motor  torque. 

At  constant  slip,  s,  the  motor  torque,  D,  is  proportional  to  the 
square  of  the  impressed  e.m.f.,  e2.  If  by  a  variation  of  slip 
caused  by  a  fluctuation  of  load  the  motor  current,  i,  varies  by  di, 
if  the  terminal  voltage,  e,  remains  constant  the  motor  torque,  Z>, 

varies  by  the  fraction  ks  =  yr  -TT-J  or  the  stability  coefficient  of 

the  motor.  If,  however,  by  the  variation  of  current,  di,  the 
impressed  e.m.f.,  e,  of  the  motor  varies,  the  motor  torque,  D, 
being  proportional  to  e2,  still  further  changes,  proportional  to 

the  change  e2,  that  is,  by  the  fraction  kr  =  —^  -JT-  =  -  j->  and  the 

e2    di        e  di 

total  change  of  motor  torque  resultant  from  a  change,  di,  of  the 
current,  i,  thus  is  ko  =  ks  +  kr. 

Hence,  if  a  momentary  fluctuation  of  current  causes  a  momen- 
tary fluctuation  of  voltage,  the  stability  coefficient  of  the  motor 
is  changed  from  ks  to  &0  =  ks  +  kr,  and  as  kr  is  negative,  the 
voltage,  6,  decreases  with  increase  of  current,  i,  the  stability 
coefficient  of  the  system  is  reduced  by  the  effect  of  voltage  regu- 
lation of  the  supply,  kr,  and  kr  thus  can  be  called  the  regulation 
coefficient  of  the  system. 

kr  =  -  -7-.  thus  represents  the  change  of  torque  produced  by 

the  momentary  voltage  change  resulting  from  a  current  change 
di  in  the  system;  hence,  is  essentially  a  characteristic  of  the 
supply  system  and  its  regulation,  but  depends  upon  the  motor 

de 
only  in  so  far  as  -p  depends  upon  the  power-factor  of  the  load. 

In  Fig.  54  is  shown  the  regulation  coefficient,  kr,  of  the  supply 
system  of  the  motor,  at  110  volts  maintained  constant  at  the 
motor  terminals,  and  an  impedance,  Z  =  0.16  +  0.8  j,  between 
motor  terminals  and  supply  e.m.f.  As  seen,  the  regulation 
coefficient  of  the  system  drops  from  a  maximum  of  about  0.03, 


INDUCTION-MOTOR  REGULATION  141 

at  no-load,  down  to  about  X).01,  and  remains  constant  at  this 
latter  value,  over  a  very  wide  range. 

The  resultant  stability  coefficient,  or  stability  co'efficient  of  the 
system  of  motor  and  supply,  fc0  =  ks  +  krj  as  shown  in  Fig.  54, 
thus  drops  from  very  high  values  at  light-load  down  to  zero  at 
the  load  at  which  the  curves,  ks  and  kr,  in  Fig.  54  intersect,  or 
at  5800  kw.,  and  there  become  negative;  that  is,  the  motor  drops 
out  of  step,  although  still  far  below  its  maximum  torque  point, 
as  indicated  by  the  arrows  in  Fig.  54. 

Thus,  at  constant  voltage  maintained  at  the  motor  terminals 
by  some  regulating  mechanism  which  is  slower  in  its  action  than 
the  retardation  of  a  motor-speed  change  by  its  mechanical 
momentum,  the  motor  behaves  up  to  5800  watts  output  in 
exactly  the  same  manner  as  if  its  terminals  were  connected 
directly  to  an  unlimited  source  of  constant  voltage  supply,  but 
at  this  point,  where  the  slip  is  only  7  per  cent,  in  the  present 
instance,  the  motor  suddenly  drops  out  of  step  without  previous 
warning,  and  comes  to  a  standstill,  while  at  inherently  constant 
terminal  voltage  the  motor  would  continue  to  operate  up  to 
7000  watts  output,  and  drop  out  of  step  at  8250  synchronous 
watts  torque  at  16  per  cent.  slip. 

By  this  phenomenon  the  maximum  torque  of  the  motor  thus 
is  reduced  from  8250  to  6300  synchronous  watts,  or  by  nearly 
25  per  cent. 

87.  If  the  voltage  regulation  of  the  supply  system  is  more 
rapid  than  the  speed  change  of  the  motor  as  retarded  by  the 
momentum  of  motor  and  load,  the  regulation  coefficient  of  the 
system  as  regards  to  the  motor  obviously  is  zero,  and  the  motor 
thus  gives  the  normal  maximum  output  and  torque.  If  the 
regulation  of  the  supply  voltage,  that  is,  the  recovery  of  the 
terminal  voltage  of  the  motor  with  a  change  of  current,  occurs  at 
about  the  same  rate  as  the  speed  of  the  motor  can  change  with 
a  change  of  load,  then  the  maximum  output  as  limited  by  the 
stability  coefficient  of  the  system  is  intermediate  between  the 
minimum  value  of  6300  synchronous  watts  and  its  normal  value 
of  8250  synchronous  watts.  The  more  rapid  the  recovery  of 
the  voltage  and  the  larger  the  momentum  of  motor  and  load, 
the  less  is  the  motor  output  impaired  by  this  phenomenon  of 
instability.  Thus,  the  loss  of  stability  is  greatest  with  hand 
regulation,  less  with  automatic  control  by  potential  regulator, 
the  more  so  the  more  rapidly  the  regulator  works ;  it  is  very  little 


142  ELECTRICAL  APPARATUS 

with  compounded  alternators,  and  absent  where  t'he  motor 
terminal  voltage  remains  constant  without  any  control  by  prac- 
tically unlimited  generator  capacity  and  absence  of  voltage  drop 
between  generator  and  motor. 

Comparing  the  stability  coefficient,  ks,  of  the  motor  load  and 
the  stability  coefficient,  kQ,  of  the  entire  system  under  the  assumed 
conditions  of  operation  of  Fig.  54,  it  is  seen  that  the  former 
intersects  the  zero  line  very  steeply,  that  is,  the  stability  remains 
high  until  very  close  to  the  maximum  torque  point,  and  the  motor 
thus  can  be  loaded  up  close  to  its  maximum  torque  without 
impairment  of  stability.  The  curve,  fc0,  however,  intersects  the 
zero  line  under  a  sharp  angle,  that  is,  long  before  the  limit  of 
stability  is  reached  in  this  case  the  stability  of  the  system  has 
dropped  so  close  to  zero  that  the  motor  may  drop  out  of  step  by 
some  momentary  pulsation.  Thus,  in  the  case  of  instability  due 
to  the  regulation  of  the  system,,  the  maximum  output  point,  as 
found  by  test,  is  not  definite  and  sharply  defined,  but  the  stability 
gradually  decreases  to  zero,  and  during  this  decrease  the  motor 
drops  out  at  some  point.  Experimentally  the  difference  between 
the  dropping  out  by  approach  to  the  limits  of  stability  of  the 
motor  proper  and  that  of  the  system  of  supply  is  very  marked 
by  the  indefiniteness  of  the  latter. 

In  testing  induction  motors  it  thus  is  necessary  to  guard 
against  this  phenomenon  by  raising  the  voltage  beyond  normal 
before  every  increase  of  load,  and  then  gradually  decrease  the 
voltages  again  to  normal. 

A  serious  reduction  of  the  overload  capacity  of  the  motor,  due 
to  the  regulation  of  the  system,  obviously  occurs  only  at  very 
high  impedance  of  the  supply  circuit;  with  moderate  impedance 
the  curve,  kr  is  much  lower,  and  the  intersection  between  kr  and 
ks  occurs  still  on  the  steep  part  of  ks,  and  the  output  thus  is  not 
materially  decreased,  but  merely  the  stability  somewhat  reduced 
when  approaching  maximum  output. 

This  phenomenon  of  the  impairment  of 'stability  of  the  induc- 
tion motor  by  the  regulation  of  the  supply  voltage  is  of  prac- 
tical importance,  as  similar  phenomena  occur  in  many  instances. 
Thus,  with  synchronous  motors  and  converters  the  regulation 
of  the  supply  system  exerts  a  similar  effect  on  the  overload 
capacity,  and  reduces  the  maximum  output  so  that  the  motor 
drops  out  of  step,  or  starts  surging,  due  to  the  approach  to  the 
stability  limit  of  the  entire  system.  In  this  case,  with  syn- 


INDUCTION-MOTOR  REGULATION  143 

chronous  motors  and  converters,  increase  of  their  field  excita- 
tion frequently  restores  their  steadiness  by  producing  leading 
currents  and  thereby  increasing  the  power-carrying  capacity 
of  the  supply  system,  while  with  surging  caused  by  instability 
of  the  synchronous  motor  the  leading  currents  produced  by 
increase  of  field  excitation  increase  the  surging,  and  lowering  the 
field  excitation  tends  toward  steadiness. 


CHAPTER  VII 
HIGHER  HARMONICS  IN  INDUCTION  MOTORS 

88.  The  usual  theory  and  calculation  of  induction  motors, 
as  discussed  in  "  Theoretical  Elements  of  Electrical  Engineer- 
ing" and  in  "  Theory  and  Calculation  of  Alternating-current 
Phenomena,"  is  based  on  the  assumption  of  the  sine  wave.  That 
is,  it  is  assumed  that  the  voltage  impressed  upon  the  motor 
per  phase,  and  therefore  the  magnetic  flux  and  the  current,  are 
sine  waves,  and  it  is  further  assumed,  that  the  distribution  of 
the  winding  on  the  circumference  of  the  armature  or  primary, 
is  sinusoidal  in  space.  While  in  most  cases  this  is  sufficiently 
the  case,  it  is  not  always  so,  and  especially  the  space  or  air-gap 
distribution  of  the  magnetic  flux  may  sufficiently  differ  from  sine 
shape,  to  exert  an  appreciable  effect  on  the  torque  at  lower 
speeds,  and  require  consideration  where  motor  action  and 
braking  action  with  considerable  power  is  required  throughout 
the  entire  range  of  speed. 

Let  then: 

e  =  ei  cos  <£  +  e3  cos  (30  —  a3)  +  e5  cos  (50  —  «5)  +  #7  cos 
(70  -  «7)  +  e9  cos  (9  0  -  a»)  +  .    .    .  (1) 

be  the  voltage  impressed  upon  one  phase  of  the  induction  motor. 
If  the  motor  is  a  quarter-phase  motor,  the  voltage  of  the 

7T 

second  motor  phase,  which  lags  90°  or  ~  behind  the  first  motor 
phase,  is: 

e'  =  ei  cos  (0  —  |j  +  e3  cos  (3  </>  --  —  —  a3j  +  e2  cos  ^5  0  —  ^  —  "*) 

+  67  COS  (7  0  --  ~  —  CiT\  +  69  COS  (9  0  --  ~  —  agj  +     .     .     . 

=  ei  cos  (0  —  ^j  +  e3  cos  ^3  0  —  «3  +  |)  +  e&  cos 

(7  0  -  on  +  ^)  +  e9  cos  (9  0  -  0:9  -  |j  +  .  .  .  (2) 


cos 


The  magnetic  flux  produced  by  these  two  voltages  thus  con- 
sists of  a  series  of  component  fluxes,  corresponding  respectively 

144 


HIGHER  HARMONICS  145 

to  the  successive  components.  The  secondary  currents  induced 
by  these  component  fluxes,  and  the  torque  produced  by  the 
secondary  currents,  thus  show  the  same  components. 

Thus  the  motor  torque  consists  of  the  sum  of  a  series  of 
components : 

The  main  or  fundamental  torque  of  the  motor,  given  by  the 
usual  sine-wave  theory  of  the  induction  motor,  and  due  to  the 
fundamental  voltage  wave: 

61  COS 

(3) 


61  COS  (  0   —  - 


is  shown  as  T\  in  Fig.  55,  of  the  usual  shape,  increasing  from 
standstill,  with  increasing  speed,  up1  to  a  maximum  torque,  and 
then  decreasing  again  to  zero  at  synchronism. 
The  third  harmonics  of  the  voltage  waves  are : 

e3  cos  (3  0  —  as), 

/  ir\  (4} 

e3 cos  (3  0  —  as  +  ~)  ' 

\  £il 

As  seen,  these  also  constitute  a  quarter-phase  system  of 
voltage,  but  the  second  wave,  which  is  lagging  in  the  funda- 
mental, is  90°  leading  in  the  third  harmonic,  or  in  other  words, 
the  third  harmonic  gives  a  backward  rotation  of  the  poles  with 
triple  frequency.  It  thus  produces  a  torque  in  opposite  direc- 
tion to  the  fundamental,  and  would  reach  its  synchronism,  that 
is,  zero  torque,  at  one-third  of  synchronism  in  negative  direction, 
or  at  the  speed  S,  =  —  M>  given  in  fraction  of  synchronous  speed. 
For  backward  rotation  above  one-third  synchronism,  this  triple 
harmonic  then  gives  an  induction  generator  torque,  and  the 
complete  torque  curve  given  by  the  third  harmonics  thus  is  as 
shown  by  curve  T*  of  Fig.  55. 

The  fifth  harmonics: 


e5  cos  (5  0  —  a 5), 
65  cos  (5  05  —  «5  —  2) 


(5) 


give  again  phase  rotation  in  the  same  direction  as  the  funda- 
mental, that  is,  motor  torque,  and  assist  the  fundamental.  But 
synchronism  is  reached  at  one-fifth  of  the  synchronous  speed  of 

the  fundamental,   or  at:  S  =   +%,  and  above  this  speed,  the 

10 


146 


ELECTRICAL  APPARATUS 


fifth  harmonic  becomes  induction  generator,  due  to  oversyn- 
chronous  rotation,  and  retards.  Its  torque  curve  is  shown  as 
Tb  in  Fig.  55. 

The  seventh  harmonic  again  gives  negative  torque,  due  to 
backward  phase  rotation  of  the  phases,  and  reaches  synchronism 
at  S  =  —  M>  that  is,  one-seventh  speed  in  backward  rotation, 
as  shown  by  curve  TI  in  Fig.  55. 


OUARTERPHASE 
INDUCTION   MOTOF 


FIG.  55. — Quarter-phase  induction  motor,  component  harmonics  and 
resultant  torque. 


The  ninth  harmonic  again  gives  positive  motor  torque  up  to 
its  synchronism,  S  =  %,  and  above  this  negative  induction 
generator  torque,  etc. 

We  then  have  the  effects  of  the  various  harmonics  on  the 


QUARTER-PHASE  INDUCTION  MOTOR 


Order  of  harmonics  

1 

3 

5 

7 

9 

11 

13 

Phase  rotation  

+ 

-f 

+ 

+ 

Synchronous  speed  :  S  =  

+  1 

-X 

+x 

-Yi 

+  X 

-Hi 

+KS 

Torque  positive  up  to  :  S  =  .  .  . 

+  1 

— 

+x 

— 

+  1A 

— 

.+K3 

otherwise  negative. 

HIGHER  HARMONICS 


147 


Adding  now  the  torque  curves  of  the  various  voltage  harmonics, 
jP3,  T*>,  TT,  to  the  fundamental  torque  curve,  TI,  of  the  induction 
motor,  gives  the  resultant  torque  curve,  T. 

As  seen  from  Fig.  55,  if  the  voltage  harmonics  are  consider- 
able, the  torque  curve  of  the  motor  at  lower  speeds,  forward 
and  backward,  that  is,  when  used  as  brake,  is  rather  irregular, 
showing  depressions  or  "dead  points." 

89.  Assume  now,  the  general  voltage  wave  (1)  is  one  of  the 
three-phase  voltages,  and  is  impressed  upon  one  of  the  phases 
of  a  three-phase  induction  motor.  The  second  and  third 

phase  then  is  lagging  by  -5-  and  -5-  respectively  behind  the  first 

o  o 

phase  (1)  : 

/  2?r\    .  /_  GTT   '        \ 

e'  =  ei  cos  \<t>  -    -_  J  +  e3  cos  (3  0  --  —  ^  «3j 

/e  10  TT          \    .  /_          14  TT          \ 

+  eb  cos  15  <£  ----  -  -  ---  abj  -f  67  cos  (7  <f>  --  -  --  «7j 

(9  <£  -  -  «9j  +    .    . 


e9  cos 


cos 


cos 


cos 


65  COS 


-   -y    +  e3  cos  (30-  «3) 

0  -  «5  -f  -  -~j  +  e7  cos 

+  eg  cos  (9  </>  —  0:9)  +    • 
-•  -jp)  +  e3  cos  (3  0  —  a3) 


I  r  ,      4r7T\      ,  /„  4r7T 

(50  -  as  +  -Q-)  4-  e7  cos (70  -  «7  -  -Q- 

\  o  /  \  o 

+  e9  cos  (9  0  -  a-9)  + 


(6) 


Thus  the  voltage  components  of  different  frequency,  impressed 
upon  the  three  motor  phases,  are: 


ei  cos  0 

63  COS 

es  cos 

C7  COS 

eg  COS 

(3  0  -  a.) 

(50  -  as) 

(70  —  en) 

(9  0  -  a») 

e\  cos 

e6  cos 

ei  cos 

(2     \ 

C3  COS 

(2     \ 

(9     \ 

C9  COS 

*  ~  ~3/ 

(3  0  -  a.) 

5  0  -  as  +  -yj 

7  0  -  a?  -J 

(9  0  -  a») 

ei  cos 

es  cos 

67  COS 

(-¥) 

es  cos 
(3  0  -  a,) 

(5*-°s  +  T) 

/                      4»\ 
1  7  0  —  a?  —  ~ 
V                        3  / 

?9  COS 

(9  0  -  a») 

Fundamental.... 

3d 

5th 

7th 

9th 

148 


ELECTRICAL  APPARATUS 


As  seen,  in  this  case  of  the  three-phase  motor,  the  third 
harmonics  have  no  phase  rotation,  but  are  in  phase  with  each 
other,  or  single-phase  voltages.  The  fifth  harmonic  gives 
backward  phase  rotation,  and  thus  negative  torque,  while  the 
seventh  harmonic  has  the  same  phase  rotation,  as  the  funda- 
mental, thus  adds  its  torque  up  to  its  synchronous  speed,  S  = 
+  H>  and  above  this  gives  negative  or  generator  torque.  The 
ninth  harmonic  again  is  single-phase. 

Fig.  56  shows  the  fundamental  'torque,  Ti,  the  higher  harmonics 


FIG.  56. — Three-phase  induction  motor,  component  harmonics  and 
resultant  torque. 

of  torque,  T&  and  TT,  and  the  resultant  torque,  T.  As  seen,  the 
distortion  of  the  torque  curve  is  materially  less,  due  to  the 
absence,  in  Fig.  56,  of  the  third  harmonic  torque. 

However,  while  the  third  harmonic  (and  its  multiples)  in  the 
three-phase  system  of  voltages  are  in  phase,  thus  give  no  phase 
rotation,  they  may  give  torque,  as  a  single-phase  induction  motor 
has  torque,  at  speed,  though  at  standstill  the  torque  is  zero. 

Fig.  57  B  shows  diagrammatically,  as  T,  the  development  of 
the  air-gap  distribution  of  a  true  three-phase  winding,  such  as 
used  in  synchronous  converters,  etc.  Each  phase  1,  2,  3,  covers 

one-third  of  the  pitch  of  a  pair  of  poles  or  -$-,  of  the  upper  layer, 


HIGHER  HARMONICS 


149 


and  its  return,  1',  2',  3',  covers  another  third  of  the  circumference 
of  two  poles,  in  the  lower  layer  of  the  armature  winding,  180° 
away  from  1,  2,  3.  However,  this  type  of  true  three-phase  wind- 
ing is  practically  never  used  in  induction  or  synchronous  machines, 
but  the  type  of  winding  is  used,  which  is  shown  as  S,  in  Fig. 
57  C.  This  is  in  reality  a  six-phase  winding:  each  of  the  three 


Q 


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FIG.  57. — Current  distribution  at  air  gap  of  induction  motor,  fundamental 

and  harmonics. 

phases,  1,  2,  3,  covers  only  one-sixth  of  the  pitch  of  a  pair  of 
poles,  or  5-  or  60°,  and  between  the  successive  phases  is  placed 

o 

the  opposite  phase,  connected  in  the  reverse  direction.  Thus 
the  return  conductors  of  phases  1,  2,  3  of  the  upper  layer,  are 
shown  in  the  lower  layer  as  ]/,  2',  3';  in  the  upper  layer,  above 
I',  2',  3',  is  placed  again  the  phase  1,  2,  3,  but  connected  in  the 
reverse  direction,  and  indicated  as  10,  20,  3o.  As  10  is  connected 
in  the  reverse  direction  to  1,  and  I'  is  the  return  of  1,  lo  is  in 


150  ELECTRICAL  APPARATUS 

phase  with  1',  and  the  return  of  10:  I'o,  is  in  the  lower  layer,  in 
phase  with,  and  beneath  1.  Thus  the  phase  rotation  is:  1,  — 3, 
2,  -1,  3,  -2,  1,  etc. 

For  comparison,  Fig.  57  A  shows  the  usual  quarter-phase 
winding,  Q,  of  the  same  general  type  as  the  winding,  Fig.  57  C. 

If  then  the  three  third  harmonics  of  1,  2  and  3  are  in  phase 
with  each  other,  for  these  third  harmonics  the  true  three-phase 
winding,  T,  gives  the  phase  diagram  shown  as  Ts  in  Fig.  57  D. 
As  seen,  the  current  flows  in  one  direction,  single-phase,  through- 
out the  entire  upper  layer,  and  in  the  opposite  direction  in  the 
lower  layer,  and  thus  its  magnetizing  action  neutralizes,  that  is, 
there  can  be  no  third  harmonic  flux  in  the  true  three-phase 
winding. 

The  third  harmonic  diagram  of  the  customary  six-phase  ar- 
rangement of  three-phase  winding,  S,  is  shown  as  83  in  Fig.  57 
E.  As  seen,  in  this  case  alternately  the  single-phase  third  har- 

7T 

monic  current  flows  in  one  direction  for  60°  or  0>  and  in  the 

o 

7T 

opposite  direction  for  the  next  «.     In  other  words,  a  single-phase 

o 

m.m.f.  and  single-phase  flux  exists,  of  three  times  as  many  poles 
as  the  fundamental  flux. 

Thus,  with  the  usual  three-phase  induction-motor  winding, 
a  third  harmonic  in  the  voltage  wave  produces  a  single-phase 
triple  harmonic  flux  of  three  times  the  number  of  motor  poles, 
and  this  gives  a  single-phase  motor-torque  curve,  that  is,  a  torque 
which,  starting  with  zero  at  standstill,  increases  to  a  maximum 
in  positive  direction  or  assisting,  and  then  decreases  again  to  zero 
at  its  synchronous  speed,  and  above  this,  becomes  negative  as 
single-phase  induction-generator  torque.  Triple  frequency  with 
three  times  the  number  of  poles  gives  a  synchronous  speed  of 
S  =  ±}^.  That  is,  the  third  harmonic  in  a  three-phase  vol- 
tage may  give  a  single-phase  motor  torque  with  a  synchronous 
speed  of  one-ninth  that  of  the  fundamental  torque,  and  in  either 
direction,  as  shown  as  T3  in  dotted  lines,  in  Fig.  56. 

As  usually  the  third  harmonic  is  absent  in  three-phase  vol- 
tages, such  a  triple  harmonic  single-phase  torque,  as  shown 
dotted  in  Fig.  56,  is  of  rare  occurrence:  it  could  occur  only  in  a 
four-wire  three-phase  system,  that  is,  system  containing  the 
three  phase- wires  and  the  neutral. 

90.  All  the  torque  components  produced  by  the  higher  har- 
monics of  the  voltage  wave  have  the  same  number  of  motor  poles 


HIGHER  HARMONICS 


151 


as  the  fundamental   (except  the  single-phase  third  harmonic 
above  discussed,  and  its  multiples,  which  have  three  times  as 


FIG.  58. — Current  and  flux  distribution  in  induction-motor  air  gap,  with 
different  types  of  windings. 

many  motor  poles),  but  a  lower  synchronous  speed,  due  to  their 
higher  frequency. 

Torque  harmonics  may  also  occur,  having  the  fundamental 


152  ELECTRICAL  APPARATUS 

frequency,  but  higher  number  of  pairs  of  poles  than  the  funda- 
mental, and  thus  lower  synchronous  speeds,  due  to  the  deviation 
of  the  space  distribution  of  the  motor  winding  from  sine. 

The  fundamental  motor  torque,  T\,  of  Figs.  55  and  56,  is  given 
by  a  sine  wave  of  voltage  and  thus  of  flux,  if  the  winding  of  each 
phase  is  distributed  around  the  circumference  of  the  motor  air 
gap  in  a  sinusoidal  manner,  as  shown  as  F  under  "Sine,"  in  Fig. 
58,  and  the  flux  distribution  of  each  phase  around  the  circum- 
ference of  the  air  gap  is  sinusoidal  also,  as  shown  as  3>  under 
"Sine,"  in  Fig.  58. 

This,  however,  is  never  the  case,  but  the  winding  is  always 
distributed  in  a  non-sinusoidal  manner. 

The  space  distribution  of  magnetizing  force  and  thus  of  flux 
of  each  phase,  along  the  circumference  of  the  motor  air  gap, 
thus  can  in  the  general  case  be  represented  by  a  trigonometric 
series,  with  co  as  space  angle,  in  electrical  degrees,  that  is,  counting 
a  pair  of  poles  as  2  IT  or  360°.  It  is  then: 

The  distribution  of  the  conductors  of  one  phase,  in  the  motor 
air  gap  : 

F  =  FQ  [  cos  co  +  a3  cos  3  co  +  a5  cos  5  co  +  a?  cos  7  co 

+  «9  cos  9  co  +  .  .  .  }  ;    (8) 

here  the  assumption  is  made,  that  all  the  harmonics  are  in  phase, 
that  is,  the  magnetic  distribution  symmetrical.  This  is  prac- 
tically always  the  case,  and  if  it  were  not,  it  would  simply  add 
phase  angle,  am,  to  the  harmonics,  the  same  as  in  paragraphs  88 
and  89,  but  would  make  no  change  in  the  result,  as  the  component 
torque  harmonics  are  independent  of  the  phase  relations  between 
the  harmonic  and  the  fundamental,  as  seen  below. 

In  a  quarter-phase  motor,  the  second  phase  is  located  90° 

or  03  =  -  displaced  in  space,  from  the  first  phase,   and  thus 

Zi 

represented  by  the  expression: 

¥'  =  Folcos  co  -       +  «3  cos  (3  co  -  ~0  +  «5  cos  (5  co  -  -~j 

(  7  co  —  ^J  +  «9  cos  n)  co  —  g-j|  +    .    .  ;  ..  1 


o 


a7  cos 


+  a7  cos  /7  co  +       +  a9  cos  (9  co  -       +    .    .    .  •  •       (9) 


HIGHER  HARMONICS  153 

Such  a  general  or  non-sinusoidal  space  distribution  of  magnetiz- 
ing force  and  thus  of  magnetic  flux,  as  represented  by  F  and  F't 
can  be  considered  as  the  superposition  of  a  series  of  sinusoidal 
magnetizing  forces  and  magnetic  fluxes: 

cos  co  0,3  cos  3co  «5  cos  5co 

/  7T\  /r»  i     ^\  ir  ^\ 

cos  I  co  — ^ )  a3  cos  (o  co  +  ~  I  05  cos  loco  —  -) 

\       z/  \          z/  \          z/ 

0,1  cos  7  co  a9  cos  9  co 

(7T\  /  7T^ 

7co  +  ^)  a9  cos  (9  co  —  7: 

Z/  \  Zii 

The  first  component: 

cos  co, 


(10) 


cos  I  co  —  » 


(10) 


gives  the  fundamental  torque  of  the  motor,  as  calculated  in  the 
customary  manner,  and  represented  by  TI  in  Figs.  55  and  56. 

The  second  component  of  space  distribution  of  magnetizing 
force : 

a3cos3  co, 

«3  cos '  v  --•  -1-  - '  • 


gives  a  distribution,  which  makes  three  times  as  many  cycles 
in  the  motor-gap  circumference,  than  (10),  that  is,  corresponds 
to  a  motor  of  three  times  as  many  poles.  This  component  of 
space  distribution  of  magnetizing  force  would  thus,  with  the 
fundamental  voltage  and  current  wave,  give  a  torque  curve 
reaching  synchronism  as  one-third  speed;  with  the  third  harmonic 
of  the  voltage  wave,  (11)  would  reach  synchronism  at  one-ninth, 
with  the  fifth  harmonic  of  the  voltage  wave  at  one-fifteenth  of 
the  normal  synchronous  speed. 

In  (11),  the  sign  of  the  second  term  is  reversed  from  that  in 
(10),  that  is,  in  (11),  the  space  rotation  is  backward  from  that 
of  (10).  In  other  words,  (11)  gives  a  synchronous  speed  of 
S  =  —  }^  with  the  fundamental  or  full-frequency  voltage  wave. 

The  third  component  of  space  distribution: 

C&5  COS  5  CO, 

(12) 


gives  a  motor  of  five  times  as  many  poles  as  (10),  but  with  same 
space  rotation  as  (10),  and  this  component  thus  would  give  a 
torque,  reaching  synchronism  at  S 


154 


ELECTRICAL  APPARATUS 


In  the  same  manner,  the  seventh  space  harmonic  gives 
S  =  —  3^7,  the  ninth  space  harmonic  S  =  +  ;H!>  etc. 

91.  As  seen,  the  component  torque  curves  of  the  harmonics 
of  the  space  distribution  of  magnetizing  force  and  magnetic 
flux  in  the  motor  air  gap,  have  the  same  characteristics  as  the 
component  torque  due  to  the  time  harmonics  of  the  impressed 
voltage  wave,  and  thus  are  represented  by  the  same  torque 
diagrams : 

Fig.  55  for  a  quarter-phase  motor, 

Fig.  56  for  a  three-phase  motor. 

Here  again,  we  see  that  the  three-phase  motor  is  less  liable 
to  irregularities  in  the  torque  curve,  caused  by  higher  harmonics, 
than  the  quarter-phase  motor  is. 

Two  classes  of  harmonics  thus  may  occur  in  the  induction 
motor,  and  give  component  torques  of  lower  synchronous  speed : 

Time  harmonics,  that  is,  harmonics  of  the  voltage  wave, 
which  are  of  higher  frequency,  but  the  same  number  of  motor 
poles,  and 

Space  harmonics,  that  is,  harmonics  in  the  air-gap  distribu- 
tion, which  are  of  fundamental  frequency,  but  of  a  higher  number 
of  motor  poles. 

Compound  harmonics,  that  is,  higher  space  harmonics  of 
higher  time  harmonics,  theoretically  exist,  but  their  torque 
necessarily  is  already  so  small,  that  they  can  be  neglected,  except 
where  they  are  intentionally  produced  in  the  design. 

We  thus  get  the  two  classes  of  harmonics,  and  their 
characteristics : 


Order  of  harmonic 

1 

3 

5 

7 

9 

11 

13 

15 

17 

Quarter-phase  motor: 
Phase  rotation  

Synchronous  speed         

+  1 

-H 

+K 

—  M 

+  X» 

-Mi 

+MS 

-Ms 

+K? 

_,.        rr  /  Frequency  
Time  H  {  .  T        .       , 
v  No.  of  poles  

p 

a/ 

5/ 

P 

7f 
p 

9/ 
P 

ii/ 
p 

13  / 
P 

p 

17/ 
p 

,,           rr  /  Frequency  
Space  H  <  .  T   M  .      , 
1  No.  of  poles  

Three-phase  motor: 
Phase  rotation 

f 
P 

/ 

3p 
0 

f 
5p 

f 

7p 

f 
0 

f 
HP 

f 
13  p 

f 

15  p 
0 

/ 

17  p 

Synchronous  speed 

-f  1 

_j/ 

+  M 

(+M27) 

-Hi 

+  M3 

-M? 

I  No.  of  poles  
Space  tf/Fre«y  
(  No  of  poles          .... 

P 
f 
P 

3/ 
(3p) 

0 

5/ 

P 

f 

7/ 
p 
f 

9/ 
(3p) 

0 

ii/ 
p 
f 
HP 

13  / 

P 

13  p 

15/ 
0 

17/ 
p 
/ 
17  p 

HIGHER  HARMONICS  155 

92.  The  space  harmonics  usually  are  more  important  than  the 
time  harmonics,  as  the  space  distribution  of  the  winding  in  the 
motor  usually  materially  differs  from  sinusoidal,  while  the  devia- 
tion of  the  voltage  wave  from  sine  shape  in  modern  electric  power- 
supply  systems  is  small,  and  the  time  harmonics  thus  usually 
negligible. 

The  space  harmonics  can  easily  be  calculated  from  the  dis- 
tribution of  the  winding  around  the  periphery  of  the  motor  air 
gap.  (See  "  Engineering  Mathematics,"  the  chapter  on  the 
trigonometric  series.) 

A  number  of  the  more  common  winding  arrangements  are 
shown  in  Fig.  58,  in  development.  The  arrangement  of  the 
conductors  of  one  phase  is  shown  to  the  left,  under  F,  and  the 
wave  shape  of  the  m.m.f.  and  thus  the  magnetic  flux  produced 
by  it  is  shown  under  $  to  the  right.  The  pitch  of  a  turn  of  the 
winding  is  indicated  under  F. 

Fig.  58  shows: 

Full-pitch  quarter-phase  winding:      Q  —  0. 

Full-pitch  six-phase  winding:  S  —  0. 

This  is  the  three-phase  winding  almost  always  used  in  induction 
and  synchronous  machines. 

Full-pitch  three-phase  winding :          T  —  0. 

This  is  the  true  three-phase  winding,  as  used  in  closed-circuit 
armatures,  as  synchronous  converters,  but  of  little  importance 
in  induction  and  synchronous  motors. 
%,  %  and  J^ -pitch  quarter-phase  windings: 

Q  -  K;  Q  -  V*\  Q  -  H- 

%,  %  and  ^-pitch  six-phase  windings: 

s  -  M;  s  -  M;  s  -  Y*. 

%-pitch  true  three-phase  windings:  T  —  %. 

As  seen,  the  pitch  deficiency,  p,  is  denoted  by  the  index. 
Denoting  the  winding,  F,  on  the  left  side  of  Fig.  58,  by  the 
Fourier  series: 

F  =  FQ  (cos  w  +  as  cos  3  co  +  a5  cos  5  co  +  a7  cos  7  co  +  .   .   . ).  (13) 
It  is,  in  general : 

IT 

FQan  =  -  f  Vcosrccodco.  (14) 

7T  JO 

If,  then:  p  =  pitch  deficiency, 

q  =  number  of  phases 


156  ELECTRICAL  APPARATUS 

(four  with  quarter-phase,  Q,  six  with  six-phase,  S}  three  with 
three-phase,  T7); 

any  fractional  pitch  winding  then  consists  of  the  superposition 
of  two  layers : 

From  co  =  0  to  co  =  — \-  —-) 
and 

from  co  =  0  to  co  =       -  ~> 

and  the  integral  (14)  become: 

v     ,    PIT  TT         Vir 


F0an  =  —     I   cos  n cod co  +   I  cos  ncodco 

7T 


fi  +  ^  CQ 

I   cos  rKjodu  -f-   I  ( 
Jo  Jo 

r4F(    .         /7T    ,    PTT\    .  fir        pir 

=  ™{S™n(q+-2)+Smn(q-^ 

SF    .    mr        pmr  ,,-, 

=  —  sm  —  cos  ^—>  (15) 

mr          q  2 


(16) 


as  for:     n  =  l',an  —  1,  it  is,  substituted  in  (15): 
8F  _  ff0 

sin  -  cos  ~ 

hence,  substituting  (16)  into  (15) : 

.    mr          pmr 
sm  —  cos 

a.=  --2- -*—  (17) 

sin  -  cos  -7:- 
q          2 

For  full-pitch  winding: 

p  =  0. 
It  is,  from  (17) : 


.     mr 
sm  — 

a.'  =  ~  -4-  (18) 

.       7T 

sm- 


HIGHER  HARMONICS  157 

and  for  a  fractional-pitch  winding  of  pitch  deficiency,  p,  it  thus  is  : 

prnr 
cos^- 

an  =  an°  -  (19) 

PIT 
cos  — 

93.  By  substituting  the  values:  q  =  4,  6,  3  and  p  —  0,  J^j, 
J£,  J^,  into  equation  (17),  we  get  the  coefficients  an  of  the 
trigonometric  series: 

F  =  Fo  {  cos  co  +  a3  cos  3  co  +  a5  cos  5  co  +  a?  cos  7  co  +  .    .    .    }  , 

(20) 

which  represents  the  current  distribution  per  phase  through  the 
air  gap  of  the  induction  machine,  shown  by  the  diagrams  F  of 
Fig.  58. 

The  corresponding  flux  distribution,  <£,  in  Fig.  58,  expressed  by 
a  trignometric  series: 

<|>  =  4>o  {  sin  co  +  63  sin  3  co  +  65  sin  5  to  +  67  sin  7  co  -|-  .    .    ,    } 

(21) 

could  be  calculated  in  the  same  manner,  from  the  constructive 
characteristics  of  $  in  Fig.  58. 

It  can,  however,  be  derived  immediately  from  the  consideration, 
that  <£  is  the  summation,  that  is,  the  integral  of  F  : 


(22) 
and  herefrom  follows: 


and  this  gives  the  coefficients,  bnj  of  the  series,  <£. 

In  the  following  tables  are  given  the  coefficients  an  and  &„, 
for  the  winding  arrangements  of  Fig.  58,  up  to  the  twenty-first 
harmonic. 

As  seen,  some  of  the  lower  harmonics  are  very  considerable 
thus  may  exert  an  appreciable  effect  on  the  motor  torque  at  low 
speeds,  especially  in  the  quarter-phase  motor. 


158 


ELECTRICAL  APPARATUS 


o;  v-  8  §  ^  8 

O   '"N  O  O   frts  O 

I    I    I    I    I    I 


ooo^ooooopooo^oo® 
od  doo'd  o'd  o 
II  ++  I  I  +  + 


§s 


s 


33 


>0  OiC          O    iO  Or^O'OO'OO.HOiooiooiooiO 

p^woo  woo  «oooooooooo  oopooo 
o'dXddXdddddo'odddo'do'dddX 
+  1  I  I  +  +  +  +  +  I  I  I  I  I  I  +  +  +  +  I  I 


OOiCOOO>OOiCOO»COO»OOO 
ICO*— (O1OO1OO*CO1C 

ooooooooooo 


o  t-oo  t-oo  t-poc 
oXoo'XddXodoooooooooooooo 

+  +  +  +  +  +I   I   I   I   I   I   I++I   I   I   I +  +  +  + 


*i       IS  £2       S  oo  co 

SB  •            oo  co 

CO           O    CO           O  T+<    O 

o    «jo-<    wo  -00oo 

.     \rH         .          •     \jH         •  •     ^" '     ^- ^ 

O^^sOO?*\O  •                OO 


o  o  o  o 

111  + 


o  "I  °. 

o  o 


8^8 


p»>  O 

I       I 


Xd 


58 


oooooooooooooooo 

+  +  +  I  I  ++  I  +  +  + I  I +  + 


o^-oooooooooooooooo 

I   I   I   I   I ++  I   I ++ I   I ++  I   I 


I-H  CO    <N 

1-H  CM     CM  ~-H 


o 

^d 


0 

^d 

1    1 


COO                   i-HCOCOi-H  (Nt^ 

000                 m3o22  C^M 

oo^poo^.  °.  >-^  o  o  o  N.  ° 

o  d            d  d  d  d  d  d 


§co 
CO 


§CD    l> 
^S2 

o  o  o 


0^0000000000000000 

I  I  I  I  I  +  +  +  +  +  +  I  I  I  I  +  + 


+  + 


o  o  o 

I  +  + 


II  II 

4?    I 


II  II 


CD       CO 


o- 


CHAPTER  VIII 
SYNCHRONIZING  INDUCTION  MOTORS 

94.  Occasionally  two  or  more  induction  motors  are  operated 
in  parallel  on  the  same  load,  as  for  instance  in  three-phase  rail- 
roading, or  when  securing  several  speeds  by  concatenation. 
In  this  case  the  secondaries  of  the  induction  motors  may  be 
connected  in  multiple  and  a  single  rheostat  used  for  starting 
and  speed  control.  Thus,  when  using  two  motors  in  concatena- 
tion for  speeds  from  standstill  to  half  synchronism,  from  half 
synchronism  to  full  speed,  the  motors  may  also  be  operated  on 
a  single  rheostat  by  connecting  their  secondaries  in  parallel. 
As  in  parallel  connection  the  frequency  of  the  secondaries  must 
be  the  same,  and  the  secondary  frequency  equals  the  slip,  it 
follows  that  the  motors  in  this  case  must  operate  at  the  same  slip, 
that  is,  at  the  same  frequency  of  rotation,  or  in  synchronism  with 
each  other.  If  the  connection  of  the  induction  motors  to  the 
load  is  such  that  they  can  not  operate  in  exact  step  with  each 
other,  obviously  separate  resistances  must  be  used  in  the  motor 
secondaries,  so  as  to  allow  different  slips.  When  rigidly  connect- 
ing the  two  motors  with  each  other,  it  is  essential  to  take  care 
that  the  motor  secondaries  have  exactly  the  same  relative  posi- 
tion to  their  primaries  so  as  to  be  in  phase  with  each  other,  just 
as  would  be  necessary  when  operating  two  alternators  in  parallel 
with  each  other  when  rigidly  connected  to  the  same  shaft  or 
when  driven  by  synchronous  motors  from  the  same  supply. 
As  in  the  induction-motor  secondary  an  e.m.f.  of  definite  fre- 
quency, that  of  slip,  is  generated  by  its  rotation  through  the 
revolving  motor  field,  the  induction-motor  secondary  is  an 
alternating-current  generator,  which  is  short-circuited  at  speed 
and  loaded  by  the  starting  rheostat  during  acceleration,  and  the 
problem  of  operating  two  induction  motors  with  their  secondaries 
connected  in  parallel  on  the  same  external  resistance  is  thus  the 
same  as  that  of  operating  two  alternators  in  parallel.  In  general, 
therefore,  it  is  undesirable  to  rigidly  connect  induction-motor 
secondaries  mechanically  if  they  are  electrically  connected  in 
parallel,  but  it  is  preferable  to  have  their  mechanical  connection 

159 


160  ELECTRICAL  APPARATUS 

sufficiently  flexible,  as  by  belting,  etc.,  so  that  the  motors  can 
drop  into  exact  step  with  each  other  and  maintain  step  by  their 
synchronizing  power. 

It  is  of  interest,  then,  to  examine  the  synchronizing  power  of 
two  induction  motors  which  are  connected  in  multiple  with 
their  secondaries  on  the  same  rheostat  and  operated  from  the 
same  primary  impressed  voltage. 

95.  Assume  two  equal  induction  motors  with  their  primaries 
connected  to  the  same  voltage  supply  and  with  their  secondaries 
connected  in  multiple  with  each  other  to  a  common  resistance, 
r,  and  neglecting  for  simplicity  the  exciting  current  and  the  vol- 
tage drop  in  the  impedance  of  the  motor  primaries  as  not  mate- 
rially affecting  the  synchronizing  power. 

Let  Zi  =  7*1  +  j%i  =  secondary  self-inductive  impedance  at 
full  frequency;  s  =  slip  of  the  two  motors,  as  fraction  of  syn- 
chronism; e0  =  absolute  value  of  impressed  voltage  and  thus, 
when  neglecting  the  primary  impedance,  of  the  voltage  generated 
in  the  primary  by  the  rotating  field. 

If  then  the  two  motor  secondaries  are  out  of  phase  with  each 
other  by  angle  2  r,  and  the  secondary  of  the  motor  1  is  behind  in 
the  direction  of  rotation  and  the '  secondary  of  the  motor  2 
ahead  of  the  average  position  by  angle  T,  then: 

EI  =  seQ  (cos  T  +  j  sin  T)  =  secondary  generated 

e.m.f.  of  the  first  motor,  (1) 

E2  =  seQ  (cos  T  —  j  sin  T)  =  secondary  generated 

e.m.f.  of  the  second  motor.  (2) 

And  if  /i  =  current  coming  from  the  first,  72  =  current  coming 
from  the  second  motor  secondary,  the  total  current,  or  current 
in  the  external  resistance,  r,  is: 

/  -  h  +  h;  (3) 

it  is  then,  in  the  circuit  comprising  the  first  motor  secondary 
and  the  rheostat,  r, 

^-/xZ-Jr^O,  (4) 

in  the  circuit  comprising  the  second  motor  secondary  and  the 
rheostat,  r, 

#2  -  h%  ~  Jr  =  0,  (5) 

where 

Z  —  TI  +  jsxi', 


SYNCHRONIZING  INDUCTION  MOTORS         161 

substituting  (3)  into  (4)  and  (5)  and  rearranging  gives: 

#!  -  A  (Z  +  r)  -  hr  =  0, 
#2  -  />  -  72  (Z  +  r)  =  0. 

These  two  equations  added  and  subtracted  give: 

El  +  Ez  -  (/!  +  I2)  (Z  +  2  r)  =  0, 
E,  -  E2  -  (/!  -  72)  Z  ^  0; 
hence, 


and 


(6) 


Substituting  for  convenience  the  abbreviations, 

(7) 
=  Qi  —  jbi 

into  equations  (6)  and  substituting  (1)  and  (2)  into  (6),  gives: 

/i  +  /2  =  2  se0Y  cos  r, 

1 1  —  /2  =  +  2jseoYismr;  (8) 

hence, 

72i  =  SeQ  { F  cos  r  +  jYl  sin  r}  (9) 

is  the  current  in  the  secondary  circuit  of  the  motor,  and  there- 
fore also  the  primary  load  current,  that  is,  the  primary  current 
corresponding  to  the  secondary  current,  and  thus,  when  neg- 
lecting the  exciting  current,  also  the  primary  motor  current, 
where  the  upper  sign  corresponds  to  the  first,  or  lagging,  the 
lower  sign  to  the  second,  or  leading,  motor. 
Substituting  in  (9)  for  F  and  FI  gives : 

/21  =  se0  { (g  cos  r  ±  bi  sin  T)  —  j  (b  cos  r  +  #1  sin  r) },    (10) 
the  primary  e.m.f.  corresponding  hereto  is: 

$2*  =  e<>  {COST  +  j  sinrj,  (11) 

where  again  the  upper  sign  corresponds  to  the  first,  the  lower  to 
the  second  motor. 

The  power  consumed  by  the  current,  /21,  with  the  e.m.f.,  $2*, 
11 


162  ELECTRICAL  APPARATUS 

is  the  sum  of  the  products  of  the  horizontal  components,  and 
of  the  vertical  components,  that  is,  of  the  real  components  and 
of  the  imaginary  components  of  these  two  quantities  (as  a 
horizontal  component  of  one  does  not  represent  any  power  with 
a  vertical  component  of  the  other  quantity,  being  in  quadrature 
therewith). 

PS  =  |&i  /.'I, 

where  the  brackets  denote  that  the  sum  of  the  product  of  the 
corresponding  parts  of  the  two  quantities  is  taken. 

As  discussed  in  the  preceding,  the  torque  of  an  induction  motor, 
in  synchronous  watts,  equals  the  power  consumed  by  the  primary 
counter  e.m.f.;  that  is: 

IV  =  IV, 

and  substituting  (10)  and  (11)  this  gives: 
IV  =  seo2  {COST  (g  COST  ±  bi  sin  T)  +  sin  T  (6  cos  T  +  07  sin  T)} 

.  seo* ei+i _ eijj? cos 2 r ± ^-JL sin 2.r  1 , 


and  herefrom  follows  the  motor  output  or  power,  by  multiplying 
with  (1-8). 

The  sum  of  the  torques  of  both  motors,  or  the  total  torque,  is : 

The  difference  of  the  torque  of  both  motors,  or  the  synchroniz- 
ing torque,  is: 

2  Ds  =  S602  (61  -  b)  sin  2  r,  (14) 

where,  by  (7), 

ri  n-2r 


m 


SXi 


,  , 

61  =  —  >  o  =  -- 

mi  m 

mi  =  n2  +  s2zi2,  w  =  (n  +  2r)2  -f 


(15) 


In  these  equations  primary  exciting  current  and  primary 
impedance  are  neglected.  The  primary  impedance  can  be  intro- 
duced in  the  equations,  by  substituting  (ri  +  sr0)  for  ri,  and 
(xi  +  XQ)  for  Xiy  in  the  expression  of  mi  and  m,  and  in  this  case 
only  the  exciting  current  is  neglected,  and  the  results  are  suffi- 
ciently accurate  for  most  purposes,  except  for  values  of  speed 


SYNCHRONIZING  INDUCTION  MOTORS        163 

very  close  to  synchronism,  where  the  motor  current  is  appreciably 
increased  by  the  exciting  current.     It  is,  then: 

mi  =  (ri  +  rs0)2  +  s2  (xi  +  x0)2, 

all  the  other  equations  remain  the  same. 
From  (15)  and  (16)  follows 

bi  -  b  _  2srxt  (n  +  sr0  +  r)  ^ 


hence,  is  always  positive. 

96.  (bi  —  b)  is  always  positive,  that  is,  the  synchronizing 
torque  is  positive  in  the  first  or  lagging  motor,  and  negative  in 
the  second  or  leading  motor;  that  is,  the  motor  which  lags  in 
position  behind  gives  more  power  and  thus  accelerates,  while  the 
motor  which  is  ahead  in  position  gives  less  power  and  thus 
drops  back.  Hence,  the  two  motor  armatures  pull  each  other 
into  step,  if  thrown  together  out  of  phase,  just  like  two  alternators. 

The  synchronizing  torque  (14)  is  zero  if  r  =  0,  as  obvious, 
as  f or  r  =  0  both  motors  are  in  step  with  each  other.  The  syn- 
chronizing torque  also  is  zero  if  r  =  90°,  that  is,  the  two  motor 
armatures  are  in  opposition.  The  position  of  opposition  is 
unstable,  however,  and  the  motors  can  not  operate  in  opposition, 
that  is,  for  r  =  90°,  or  with  the  one  motor  secondary  short- 
circuiting  the  other;  in  this  position,  any  decrease  of  r  below 
90°  produces  a  synchronizing  torque  which  pulls  the  motors 
together,  to  r  =  0,  or  in  step.  Just  as  with  alternators,  there 
thus  exist  two  positions  of  zero  synchronizing  power — with  the 
motors  in  step,  that  is,  their  secondaries  in  parallel  and  in  phase, 
and  with  the  motors  in  opposition,  that  is,  their  secondaries  in 
opposition — and  the  former  position  is  stable,  the  latter  unstable, 
and  the  motors  thus  drop  into  and  retain  the  former  position, 
that  is,  operate  in  step  with  each  other,  within  the  limits  of  their 
synchronizing  power. 

If  the  starting  rheostat  is  short-circuited,  or  r  =  0,  it  is,  by 
(15),  61  =  6,  and  the  synchronizing  power  vanishes,  as  is  obvious, 
since  in  this  case  the  motor  secondaries  are  short-circuited  and 
thus  independent  of  each  other  in  their  frequency  and  speed. 

With  parallel  connection  of  induction-motor  armatures  a  syn- 
chronizing power  thus  is  exerted  between  the  motors  as  long 
as  any  appreciable  resistance  exists  in  the  external  circuit,  and 


164 


ELECTRICAL  APPARATUS 


the  motors  thus  tend  to  keep  in  step  until  the  common  starting 
resistance  is  short-circuited  and  the  motors  thereby  become  inde- 
pendent, the  synchronizing  torque  vanishes,  and  the  motors  can 
slip  against  each  other  without  interference  by  cross-currents. 

Since  the  term  — •= —  contains  the  slip,  s,  as  factor,  the  syn- 
chronizing torque  decreases  with  increasing  approach  to  syn- 
chronous speed. 


e  0=iooo  VOLTS 
z,=z0=i-J 

=  0;0.75,  2,4.5 


0.1  0.2          0.3  0.4  0.5  0.6          0.7  0.8          0.9          1.0 

FIG.  59. — Synchronizing  induction  motors :  motor  torque  and  synchronizing 

torque. 

For  r  =  0,  or  with  the  motors  in  step  with  each  other,  it  is,  by 
(12),  (15),  and  (16): 

W  =  se^  =    r   +  sr  *+2r?±$x   +x    *>         (18) 

that  is,  the  same  value  as  found  for  a  single  motor.  (As  the 
resistance  r  is  common  to  both  motors,  for  each  motor  it  enters 
as  2  r.) 

For  T  =  90°,  or  the  unstable  positions  of  the  motors,  it  is : 


(r. 
that  is,  the  same  value  as  the  motor  would  give  with  short- 


SYNCHRONIZING  INDUCTION  MOTORS        165 

circuited  armature.     This  is  to  be  expected,  as  the  two  motor 
armatures  short-circuit  each  other. 

The  synchronizing  torque  is  a  maximum  for  T  =  45°,  and  is, 
by  (14),  (15),  and  (16): 

Da  =  seo2  b^~-  (20) 

As  instances  are  shown,  in  Fig.  59,  the  motor  torque,  from 
equation  (18),  and  the  maximum  synchronizing  torque,  from 
equation  (20),  for  a  motor  of  5  per  cent,  drop  of  speed  at  full- 
load  and  very  high  overload  capacity  (a  maximum  power  nearly 
two  and  a  half  times  and  a  maximum  torque  somewhat  over 
three  times  the  rated  value),  that  is,  of  low  reactance,  as  can  be 
produced  at  low  frequency,  and  is  desirable  for  intermittent 
service,  hence  of  the  constants: 

Zi  =  Z0  =  1  +  j, 
Y  =  0.005  -  0.02  j, 

€Q   =    1000  VOltS, 

for  the  values  of  additional  resistance  inserted  into  the  armatures : 

r  =  0;  0.75;  2;  4.5, 
giving  the  values: 

1  l  +  2r 

gi       ml  ~^T 

9   Q  <JT, 

&*J    O                                                                                                                               1                        O«4/l 
1  =  ,  0  =  , 

mi  m 

mi  =  (1  +  s)2  +  4  8*t  m  =  (i  +  8  +  2  r)2  +  4  s2. 

As  seen,  in  this  instance  the  synchronizing  torque  is  higher 
than  the  motor  torque  up  to  half  speed,  slightly  below  the  motor 
torque  between  half  speed  and  three-quarters  speed,  but  above 
three-quarters  speed  rapidly  drops,  due  to  the  approach  to  syn- 
chronism, and  becomes  zero  when  the  last  starting  resistance 
is  cut  out. 


CHAPTER  IX 
SYNCHRONOUS  INDUCTION  MOTOR 

97.  The  typical  induction  motor  consists  of  one  or  a  number 
of  primary  circuits  acting  upon  an  armature  movable  thereto, 
which  contains  a  number  of  closed  secondary  circuits,  displaced 
from  each  other  in  space  so  as  to  offer  a  resultant  closed  secondary 
circuit  in  any  direction  and  at  any  position  of  the  armature  or 
secondary,  with  regards  to  the  primary  system.  In  consequence 
thereof  the  induction  motor  can  be  considered  as  a  transformer, 
having  to  each  primary  circuit  a  corresponding  secondary  cir- 
cuit— a  secondary  coil,  moving  out  of  the  field  of  the  primary 
coil,  being  replaced  by  another  secondary  coil  moving  into  the 
field. 

In  such  a  motor  the  torque  i's  zero  at  synchronism,  positive 
below,  and  negative  above,  synchronism. 

If,  however,  the  movable  armature  contains  one  closed  cir- 
cuit only,  it  offers  a  closed  secondary  circuit  only  in  the  direc- 
tion of  the  axis  of  the  armature  coil,  but  no  secondary  circuit  at 
right  angles  therewith.  That  is,  with  the  rotation  of  the  arma- 
ture the  secondary  circuit,  corresponding  to  a  primary  circuit, 
varies  from  short-circuit  at  coincidence  of  the  axis  of  the  arma- 
ture coil  with  the  axis  of  the  primary  coil,  to  open-circuit  in 
quadrature  therewith,  with  the  periodicity  of  the  armature 
speed.  That  is,  the  apparent  admittance  of  the  primary  circuit 
varies  periodically  from  open-circuit  admittance  to  the  short- 
circuited  transformer  admittance. 

At  synchronism  such  a  motor  represents  an  electric  circuit  of 
an  admittance  varying  with  twice  the  periodicity  of  the  primary 
frequency,  since  twice  per  period  the  axis  of  the  armature  coil 
and  that  of  the  primary  coil  coincide.  A  varying  admittance 
is  obviously  identical  in  effect  with  a  varying  reluctance,  which 
will  be  discussed  in  the  chapter  on  reaction  machines.  That 
is,  the  induction  motor  with  one  closed  armature  circuit  is,. at 
synchronism,  nothing  but  a  reaction  machine,  and  consequently 
gives  zero  torque  at  synchronism  if  the  maxima  and  minima  of 
the  periodically  varying  admittance  coincide  with  the  maximum 

166 


SYNCHRONOUS  INDUCTION  MOTOR  167 

and  zero  values  of  the  primary  circuit,  but  gives  a  definite  torque 
if  they  are  displaced  therefrom.  This  torque  may  be  positive 
or  negative  according  to  the  phase  displacement  between  ad- 
mittance and  primary  circuit;  that  is,  the  lag  or  lead  of  the 
maximum  admittance  with  regard  to  the  primary  maximum. 
Hence  an  induction  motor  with  single-armature  circuit  at  syn- 
chronism acts  either  as  motor  or  as  alternating-current  generator 
according  to  the  relative  position  of  the  armature  circuit  with 
respect  to  the  primary  circuit.  Thus  it  can  be  called  a  syn- 
chronous induction  motor  or  synchronous  induction  generator, 
since  it  is  an  induction  machine  giving  torque  at  synchronism. 

Power-factor  and  apparent  efficiency  of  the  synchronous  in- 
duction motor  as  reaction  machine  are  very  low.  Hence  it  is 
of  practical  application  only  in  cases  where  a  small  amount  of 
power  is  required  at  synchronous  rotation,  and  continuous  current* 
for  field  excitation  is  not  available. 

The  current  produced  in  the  armature  of  the  synchronous 
induction  motor  is  of  double  the  frequency  impressed  upon  the 
primary. 

Below  and  above  synchronism  the  ordinary  induction  motor, 
or  induction  generator,  torque  is  superimposed  upon  the  syn- 
chronous-induction machine  torque.  Since  with  the  frequency 
of  slip  the  relative  position  of  primary  and  of  secondary  coil 
changes,  the  synchronous-induction  machine  torque  alternates 
periodically  with  the  frequency  of  slip.  That  is,  upon  the  con- 
stant positive  or  negative  torque  below  or  above  synchronism 
an  alternating  torque  of  the  frequency  of  slip  is  superimposed, 
and  thus  the  resultant  torque  pulsating  with  a  positive  mean 
value  below,  a  negative  mean  value  above,  synchronism. 

When  started  from  rest,  a  synchronous  induction  motor  will 
accelerate  like  an  ordinary  single-phase  induction  motor,  but 
not  only  approach  synchronism,  as  the  latter  does,  but  run  up 
to  complete  synchronism  under  load.  When  approaching  syn- 
chronism it  makes  definite  beats  with  the  frequency  of  slip,  which 
disappear  when  synchronism  is  reached. 


CHAPTER  X 
HYSTERESIS  MOTOR 

98.  In  a  revolving  magnetic  field,  a  circular  iron  disk,  or 
iron  cylinder  of  uniform  magnetic  reluctance  in  the  direction  of 
the  revolving  field,  is  set  in  rotation,  even  if  subdivided  so  as  to 
preclude  the  production  of  eddy  currents.  This  rotation  is -due 
to  the  effect  of  hysteresis  of  the  revolving  disk  or  cylinder,  and 
such  a  motor  may  thus  be  called  a  hysteresis  motor. 

Let  I  be  the  iron  disk  exposed  to  a  rotating  magnetic  field 
.or  resultant  m.m.f.  The  axis  of  resultant  magnetization  in  the 
disk,  /,  does  not  coincide  with  the  axis  of  the  rotating  field,  but 
lags  behind  the  latter,  thus  producing  a  couple.  That  is,  the 
component  of  magnetism  in  a  direction  of  the  rotating  disk,  /, 
ahead  of  the  axis  of  rotating  m.m.f.,  is  rising,  thus  below,  and 
in  a  direction  behind  the  axis  of  rotating  m.m.f.  decreasing,  that 
is,  above  proportionality  with  the  m.m.f.,  in  consequence  of  the 
lag  of  magnetism  in  the  hysteresis  loop,  and  thus  the  axis  of 
resultant  magnetism  in  the  iron  disk,  7,  does  not  coincide  with 
the  axis  of  rotating  m.m.f.,  but  is  shifted  backward  by  an  angle, 
a,  which  is  the  angle  of  hysteretic  lead. 

The  induced  magnetism  gives  with  the  resultant  m.m.f.  a 
mechanical  couple: 

D  =  mSQ  sin  a, 
where 

$  =  resultant  m.m.f., 
<£  =  resultant  magnetism, 
a  =  angle  of  hysteretic  advance  of  phase, 
m  =  a  constant. 

The  apparent  or  volt-ampere  input  of  the  motor  is : 
p  =  m$$. 

Thus  the  apparent  torque  efficiency: 

P 

Q  =  Sm  a> 
where 

Q  =  volt-ampere  input, 
168 


HYSTERESIS  MOTOR  169 

and  the  power  of  the  motor  is: 

p  =  (l  -  s)  D  =  (1  -  s)  wfffc  sin  a, 
where 

s  =  slip  as  fraction  of  synchronism. 

The  apparent  efficiency  is: 

-Q  =  (1  -  s)sin  a. 

Since  in  a  magnetic  circuit  containing  an  air  gap  the  angle, 
a,  is  small,  a  few  degrees  only,  it  follows  that  the  apparent 
efficiency  of  the  hysteresis  motor  is  low,  the  motor  consequently 
unsuitable  for  producing  large  amounts  of  mechanical  power. 

From  the  equation  of  torque  it  follows,  however,  that  at 
constant  impressed  e.m.f.,  or  current  —  that  is,  constant  $  — 
the  torque  is  constant  and  independent  of  the  speed;  and  there- 
fore such  a  motor  arrangement  is  suitable,  and  occasionally  used 
as  alternating-current  meter. 
For  s<0,  we  have  a  <  0, 

and  the  apparatus  is  an  hysteresis  generator. 

99.  The  same  result  can  be  reached  from  a  different  point 
of  view.  In  such  a  magnetic  system,  comprising  a  movable 
iron  disk,  7,  of  uniform  magnetic  reluctance  in  a  revolving 
field,  the  magnetic  reluctance  —  and  thus  the  distribution  of 
magnetism  —  is  obviously  independent  of  the  speed,  and  conse- 
quently the  current  and  energy  expenditure  of  the  impressed 
m.m.f.  independent  of  the  speed  also.  If,  now: 

V  =  volume  of  iron  of  the  movable  part, 

(B  =  magnetic  density, 
and 

17  =  coefficient  of  hysteresis, 

the  energy  expended  by  hysteresis  in  the  movable  disk,  /,  is 
per  cycle: 


hence,  if  /  =  frequency,  the  power  supplied  by  the  m.m.f.  to 
the  rotating  iron  disk  in  the  hysteretic  loop  of  the  m.m.f.  is: 

Po  =  /TfyB1-6. 

At  the  slip,  sf,  that  is,  the  speed  (1  —  s)  /,  the  power  expended 
by  hysteresis  in  the  rotating  disk  is,  however: 

Pi  = 


170  ELECTRICAL  APPARATUS 

Hence,  in  the  transfer  from  the  stationary  to  the  revolving 
member  the  magnetic  power: 

P  =  P0  -  Pi  =  (1  -  s)/Frj(B1-6, 

has  disappeared,  and  thus  reappears  as  mechanical  work,  and 
the  torque  is: 

p 

T)  —  f —  y^mi.e 

(1  -  «)/  " 
that  is,  independent  of  the  speed. 

Since,  as  seen  in  "  Theory  and  Calculation  of  Alternating-cur- 
rent Phenomena,"  Chapter  XII,  sin  a  is  the  ratio  of  the  energy 
of  the  hysteretic  loop  to  the  total  apparent  energy  of  the  mag- 
netic cycle,  it  follows  that  the  apparent  efficiency  of  such  a  motor 
can  never  exceed  the  value  (1  —  s)  sin  a,  or  a  fraction  of  the 
primary  .hysteretic  energy. 

The  primary  hysteretic  energy  of  an  induction  motor,  as  repre- 
sented by  its  conductance,  g,  being  a  part  of  the  loss  in  the 
motor,  and  thus  a  very  small  part  of  its  output  only,  it  follows 
that  the  output  of  a  hysteresis  motor  is  a  small  fraction  only  of 
the  output  which  the  same  magnetic  structure  could  give  with 
secondary  short-circuited  winding,  as  regular  induction  motor. 

As  secondary  effect,  however,  the  rotary  effort  of  the  magnetic 
structure  as  hysteresis  motor  appears  more  or  less  in  all  induction 
motors,  although  usually  it  is  so  small  as  to  be  neglected. 

However,  with  decreasing  size  of  the  motor,  the  torque  of  the 
hysteresis  motor  decreases  at  a  lesser  rate  than  that  of  the  in- 
duction motor,  so  that  for  extremely  small  motors,  the  torque 
as  hysteresis  motor  is  comparable  with  that  as  induction  motor. 

If  in  the  hysteresis  motor  the  rotary  iron  structure  has  not 
uniform  reluctance  in  all  directions — but  is,  for  instance,  bar- 
shaped  or  shuttle-shaped — on  the  hysteresis-motor  effect  is 
superimposed  the  effect  of  varying  magnetic  reluctance,  which 
tends  to  bring  the  motor  to  synchronism,  and  maintain  it 
therein,  as  shall  be  more  fully  investigated  under  "  Reaction 
Machine"  in  Chapter  XVI. 

100.  In  the  hysteresis  motor,  consisting  of  an  iron  disk  of 
uniform  magnetic  reluctance,  which  revolves  in  a  uniformly 
rotating  magnetic  field,  below  synchronism,  the  magnetic  flux 
rotates  in  the  armature  with  the  frequency  of  slip,  and  the 
resultant  line  of  magnetic  induction  in  the  disk  thus  lags,  in 
space,  behind  the  synchronously  rotating  line  of  resultant  m.m.f. 


HYSTERESIS  MOTOR  171 

of  the  exciting  coils,  by  the  angle  of  hysteretic  lead,  a,  which  is 
constant,  and  so  gives,  at  constant  magnetic  flux,  that  is,  con- 
stant impressed  e.m.f.,  a  constant  torque  and  a  power  propor- 
tional to  the  speed. 

Above  synchronism,  the  iron  disk  revolves  faster  than  the 
rotating  field,  and  the  line  of  resulting  magnetization  in  the  disk 
being  behind  the  line  of  m.m.f.  with  regard  to  the  direction  of 
rotation  of  the  magnetism  in  the  disk,  therefore  is  ahead  of  it  in 
space,  that  is,  the  torque  and  therefore  the  power  reverses  at 
synchronism,  and  above  synchronism  the  apparatus  is  an 
hysteresis  generator,  that  is,  changes  at  synchronism  from  motor 
to  generator.  At  synchronism  such  a  disk  thus  can  give  me- 
chanical power  as  motor,  with  the  line  of  induction  lagging,  or 
give  electric  power  as  generator,  with  the  line  of  induction 
leading  the  line  of  rotation  m.m.f. 

Electrically,  the  power  transferred  between  the  electric  cir- 
cuit and  the  rotating  disk  is  represented  by  the  hysteresis  loop. 
Below  synchronism  the  hysteresis  loop  of  the  electric  circuit 
has  the  normal  shape,  and  of  its  constant  power  a  part,  propor- 
tional to  the  slip,  is  consumed  in  the  iron,  the  other  part,  pro- 
portional to  the  speed,  appears  as  mechanical  power.  At  syn- 
chronism the  hysteresis  loop  collapses  and  reverses,  and  above 
synchronism  the  electric  supply  current  so  traverses  the  normal 
hysteresis  loop  in  reverse  direction,  representing  generation  of 
electric  power.  The  mechanical  power  consumed  by  the 
hysteresis  generator  then  is  proportional  to  the  speed,  and  of 
this  power  a  part,  proportional  to  the  slip  above  synchronism, 
is  consumed  in  the  iron,  the  other  part  is  constant  and  appears 
as  electric  power  generated  by 'the  apparatus  in  the  inverted 
hysteresis  loop. 

This  apparatus  is  of  interest  especially  as  illustrating  the 
difference  between  hysteresis  and  molecular  magnetic  friction: 
the  hysteresis  is  the  power  represented  by  the  loop  between 
magnetic  induction  and  m.m.f.  or  the  electric  power  in  the 
circuit,  and  so  may  be  positive  or  negative,  or  change  from  the 
one  to  the  other,  as  in  the  above  instance,  while  molecular  mag- 
netic friction  is  the  power  consumed  in  the  magnetic  circuit  by 
the  reversals  of  magnetism.  Hysteresis,  therefore,  is  an  electrical 
phenomenon,  and  is  a  measure  of  the  molecular  magnetic  fric- 
tion only  if  there  is  no  other  source  or  consumption  of  power  in 
the  magnetic  circuit. 


CHAPTER  XI 

ROTARY  TERMINAL  SINGLE-PHASE  INDUCTION 
MOTOR 

101.  A  single-phase  induction  motor,  giving  full  torque  at 
starting  and  at  any  intermediate  speed,  by  means  of  leading  the 
supply  current  into  the  primary  motor  winding  through  brushes 
moving  on  a  segmental  commutator  connected  to  the  primary 


FIG.  60. — Diagram  of  rotary  terminal  single-phase  induction  motor. 

winding,  was  devised  and  built  by  R.  Eickemeyer  in  1891,  and 
further  work  thereon  done  later  in  Germany,  but  never  was 
brought  into  commercial  use. 

Let,  in  Fig.  60,  P  denote  the  primary  stator  winding  of  a  single- 
phase  induction  motor,  S  the  revolving  squirrel-cage  secondary 
winding.  The  primary  winding  is  arranged  as  a  ring  (or  drum) 
winding  and  connected  to  a  stationary  commutator,  C.  The 
single-phase  supply  current  is  led  into  the  primary  winding,  P, 
through  two  brushes  bearing  on  the  two  (electrically)  opposite 

172 


SINGLE-PHASE  INDUCTION  MOTOR  173 

points  of  the  commutator,  C.  These  brushes,  B,  are  arranged  so 
that  they  can  be  revolved. 

With  the  brushes,  B,  at  standstill  on  the  stationary  commutator, 
C,  the  rotor,  S,  has  no  torque,  and  the  current  in  the  stator,  P,  is 
the  usual  large  standstill  current  of  the  induction  motor.  If  now 
the  brushes,  B,  are  revolved  at  synchronous  speed,  /,  in  the  direc- 
tion shown  by  the  arrow,  the  rotor,  S,  again  has  no  torque,  but 
the  stator,  P,  carries  only  the  small  exciting  current  of  the  motor, 
and  the  electrical  conditions  in  the  motor  are  the  same,  as  would 
be  with  stationary  brushes,  B,  at  synchronous  speed  of  the  rotor, 
S.  If  now  the  brushes,  5,  are  slowed  down  below  synchronism, 
/,  to  speed,  /i,  the  rotor,  S,  begins  to  turn,  in  reverse  direction,  as 
shown  by  the  arrow,  at  a  speed,  /2,  and  a  torque  corresponding 
to  the  slip,  s  =  f  —  (/i  +  /2). 

Thus,  if  the  load  on  the  motor  is  such  as  to  require  the  torque 
given  at  the  slip,  s,  this  load  is  started  and  brought  up  to  full 
speed,  f  —  s,  by  speeding  the  brushes,  B,  up  to  or  near  synchronous 
speed,  and  then  allowing  them  gradually  to  come  to  rest  :  at  brush 
speed,  /i  =  /  —  s,  the  rotor  starts,  and  at  decreasing,  /i,  accelr- 
ates  with  the  speed  /2  =  /  —  s  —  /i,  until,  when  the  brushes 
come  to  rest  :  /i  =  0,  the  rotor  speed  is  /2  =  /  —  s. 

As  seen,  the  brushes  revolve  on  the  commutator  only  in  start- 
ing and  at  intermediate  speeds,  but  are  stationary  at  full  speed. 
If  the  brushes,  By  are  rotated  at  oversynchronous  speed:  /i>/, 
the  motor  torque  is  reversed,  and  the  rotor  turns  in  the  same 
direction  as  the  brushes.  In  general,  it  is: 


where 

/i  =  brush  speed, 
/2  =  motor  speed, 

s  =  slip  required  to  give  the  desired  torque, 

/  =  supply  frequency. 

102.  An  application  of  this  type  of  motor  for  starting  larger 
motors  under  power,  by  means  of  a  small  auxiliary  motor,  is 
shown  diagrammatically,  in  section,  in  Fig.  61. 

Po  is  the  stationary  primary  or  stator,  So  the  revolving  squirrel- 
cage  secondary  of  the  power  motor.  The  stator  coils  of  P0 
connect  to  the  segments  of  the  stationary  commutator,  Co, 
which  receives  the  single-phase  power  current  through  the 
brushes,  B0. 


174 


ELECTRICAL  APPARATUS 


These  brushes,  5o,  are  carried  by  the  rotating  squirrel-cage 
secondary,  Si,  of  a  small  auxiliary  motor.  The  primary  of  this, 
Pi,  is  mounted  on  the  power  shaft,  A,  of  the  main  motor,  and 
carries  the  commutator,  Ci,  which  receives  current  from  the 
brushes,  Bi. 

These  brushes  are  speeded  up  to  or  near  synchronism  by  some 
means,  as  hand  wheel,  H ,  and  gears,  G,  and  then  allowed  to  slow 
down.  Assuming  the  brushes  were  rotating  in  counter-clock- 
wise direction.  Then,  while  they  are  slowing  down,  the  (ex- 
ternal) squirrel-cage  rotor,  /Si,  of  the  auxiliary  motor  starts  and 


>o 

777ft 

OQ 

\ 

\ 

^ 

n 

^a 

0 

65 

linn 

p™ 

r. 

1 

1 

rn 

PI 

mm, 

mm 

rw^ 

FIG.  61. — Rotary  terminal  sngle-phase  induction  motor  with  controlling 

motor. 

speeds  up,  in  clockwise  direction,  and  while  the  brushes,  B\, 
come  to  rest,  S\  comes  up  to  full  speed,  and  thereby  brings  the 
brushes,  BQ,  of  the  power  motor  up  to  speed  in  clockwise  rotation. 
As  soon  as  BQ  has  reached  sufficient  speed,  the  power  motor  gets 
torque  and  its  rotor,  S0,  starts,  in  counter-clockwise  rotation. 
As  So  carries  Pi,  with  increasing  speed  of  SQ  and  PI,  Si  and  with 
it  the  brushes,  B0,  slow  down,  until  full  speed  of  the  power  motor, 
$o,  is  reached,  the  brushes,  B0,  stand  still,  and  the  brushes,  B\, 
by  their  friction  on  the  commutator,  Ci,  revolve  together  with 
Ci,  Pi  and  S0. 

In "  whichever   direction   the   brushes,  BI,  are  started,  in  the 
same  direction  starts  the  main  motor,  S0. 


SINGLE-PHASE  INDUCTION  MOTOR  175 

If  by  overload  the  main  motor,  So,  drops  out  of  step  and  slows 
down,  the  slowing  down  of  P\  starts  Si,  and  with  it  the  brushes, 
Bo,  at  the  proper  differential  speed,  and  so  carries  full  torque 
down  to  standstill,  that  is,  there  is  no  actual  dropping  out  of 
the  motor,  but  merely  a  slowing  down  by  overload. 

The  disadvantage  of  this  motor  type  is  the  sparking  at  the 
commutator,  by  the  short-circuiting  of  primary  coils  during  the 
passage  of  the  brush  from  segment  to  segment.  This  would 
require  the  use  of  methods  of  controlling  the  sparking,  such  as 
used  in  the  single-phase  commutator  motors  of  the  series  type, 
etc.  It  was  the  difficulty  of  controlling  the  sparking,  which 
side-tracked  this  type  of  motor  in  the  early  days,  and  later,  with 
the  extensive  introduction  of  polyphase  supply,  the  single-phase 
motor  problem  had  become  less  important. 


CHAPTER  XII 

FREQUENCY  CONVERTER  OR  GENERAL  ALTERNATING  - 
CURRENT  TRANSFORMER 

103.  In  general,  an  alternating-current  transformer  consists  of 
a  magnetic  circuit,  interlinked  with  two  electric  circuits  or  sets 
of  electric  circuits,  the  primary  circuit,  in  which  power,  sup- 
plied by  the  impressed  voltage,  is  consumed,  and  the  secondary 
circuit,  in  which  a  corresponding  amount  of  electric  power 
is  produced;  or  in  other  words,  power  is  transferred  through 
space,  by  magnetic  energy,  from  primary  to  secondary  circuit. 
This  power  finds  its  mechanical  equivalent  in  a  repulsive  thrust 
acting  between  primary  and  secondary  conductors.  Thus,  if 
the  secondary  is  not  held  rigidly,  with  regards  to  the  primary, 
it  will  be  repelled  and  move.  This  repulsion  is  used  in  the 
constant-current  transformer  for  regulating  the  current  for 
constancy  independent  of  the  load.  In  the  induction  motor, 
this  mechanical  force  is  made  use  of  for  doing  the  work:  the 
induction  motor  represents  an  alternating-current  transformer, 
in  which  the  secondary  is  mounted  movably  with  regards  to 
the  primary,  in  such  a  manner  that,  while  set  in  motion,  it  still 
remains  in  the  primary  field  -of  force.  This  requires,  that  the 
induction  motor  field  is  not  constant  in  one  direction,  but  that 
a  magnetic  field  exists  in  every  direction,  in  other  words  that 
the  magnetic  field  successively  assumes  all  directions,  as  a  so- 
called  rotating  field. 

The  induction  motor  and  the  stationary  transformer  thus  are 
merely  two  applications  of  the  same  structure,  the  former  using 
the  mechanical  thrust,  the  latter  only  the  electrical  power 
transfer,  and  both  thus  are  special  cases  of  what  may  be  called 
the  " general  alternating-current  transformer,"  in  which  both, 
power  and  mechanical  motion,  are  utilized. 

The  general  alternating-current  transformer  thus  consists  of 
a  magnetic  circuit  interlinked  with  two  sets  of  electric  circuits, 
the  primary  and  the  secondary,  which  are  mounted  rotatably 
with  regards  to  each  other.  It  transforms  between  primary 
electrical  and  secondary  electrical  power,  and  also  between 

176 


FREQUENCY  CONVERTER         177 

electrical  and  mechanical  power.  As  the  frequency  of  the  re- 
volving secondary  is  the  frequency  of  slip,  thus  differing  from 
the  primary,  it  follows,  that  the  general  alternating-current 
transformer  changes  not  only  voltages  and  current,  but  also 
frequencies,  and  may  therefore  be  called  "  frequency  converter." 
Obviously,  it  may  also  change  the  number  of  phases. 

Structurally,  frequency  converter  and  induction  motor  must 
contain  an  air  gap  in  the  magnetic  circuit,  to  permit  movability 
between  primary  and  secondary,  and  thus  they  require 'a  higher 
magnetizing  current  than  the  closed  magnetic  circuit  stationary 
transformer,  and  this  again  results  in  general  in  a  higher  self- 
inductive  impedance.  Thus,  the  frequency  converter  and  in- 
duction motor  magnetically  represent  transformers  of  high  ex- 
citing admittance  and  high  self-inductive  impedance. 

104.  The  mutual  magnetic  flux  of  the  transformer  is  pro- 
duced by  the  resultant  m.m.f.  of  both  electric  circuits.  It  is 
determined  by  the  counter  e.m.f.,  the  number  of  turns,  and  the 
frequency  of  the  electric  circuit,  by  the  equation : 

I     E  =  v^Tr/ViSlO-8, 
where 

E  =  effective  e.m.f., 
/  =  frequency, 
n  =  number  of  turns, 
$  =  maximum  magnetic  flux. 

The  m.m.f.  producing  this  flux,  or  the  resultant  m.m.f.  of 
primary  and  secondary  circuit,  is  determined  by  shape  and 
magnetic  characteristic  of  the  material  composing  the  magnetic 
circuit,  and  by  the  magnetic  induction.  At  open  secondary 
circuit,  this  m.m.f.  is  the  m.m.f.  of  the  primary  current,  which 
in  this  case  is  called  the  exciting  current,  and  consists  of  a 
power  component,  the  magnetic  power  current,  and  a  reactive 
component,  the  magnetizing  current. 

In  the  general  alternating-current  transformer,  where  the 
secondary  is  movable  with  regard  to  the  primary,  the  rate  of 
cutting  of  the  secondary  electric  circuit  with  the  mutual  mag- 
netic flux  is  different  from  that  of  the  primary.  Thus,,  the  fre- 
quencies of  both  circuits  are  different,  and  the  generated  e.m.fs. 
are  not  proportional  to  the  number  of  turns  as  in  the  stationary 

transformer,  but  to  the  product  of  number  of  turns  into  frequency. 
12 


178  ELECTRICAL  APPARATUS 

105.  Let,  in  a  general  alternating-current  transformer: 


,.     secondary  r 

=  ratio  — -. —    -*•  frequency,  or     slip   ; 
primary 


thus,  if: 


/  =  primary  frequency,  or  frequency  of  impressed  e.m.f., 
sf  =  secondary  frequency; 

and  the  e.m.f.  generated  per  secondary  turn  by  the  mutual  flux 
has  to  the  e.m.f.  generated  per  primary  turn  the  ratio,  s, 

s  =  0  represents  synchronous  motion  of  the  secondary; 

s  <  0  represents  motion  above  synchronism — driven  by  external 

mechanical  power,  as  will  be  seen; 
s  =  I  represents  standstill; 
s  >  I  represents  backward  motion  of  the  secondary, 

that  is,   motion  against  the  mechanical  force  acting  between 
primary  and  secondary  (thus  representing  driving  by  external 
mechanical  power). 
Let: 

n0  —  number  of  primary  turns  in  series  per  circuit; 
n\  =  number  of  secondary  turns  in  series  per  circuit; 

a  =  —  =  ratio  of  turns; 

ni 

Y  =  g  —  jb  =  primary  exciting  admittance  per  circuit; 

where : 

g  =  effective  conductance; 
b  =  susceptance; 

Zo  =  TO  +  jx0  =  internal  primary  self-inductive  impedance 
per  circuit, 

where : 

r0  =  effective  resistance  of  primary  circuit; 
0*0  =  self-inductive  reactance  of  primary  circuit; 
Zn  =  n   +  jxi   =   internal  secondary  self-inductive  im- 
pedance per  circuit  at  standstill,  or  for  s  =  1, 

where : 

7*1  =  effective  resistance  of  secondary  coil; 
#i  =  self-inductive  reactance  of  secondary  coil  at  stand- 
still, or  full  frequency,  s  =  1. 


FREQUENCY  CONVERTER          179 

Since  the  reactance  is  proportional  to  the  frequency,  at  the 
slip,  s,  or  the  secondary  frequency,  s/,  the  secondary  impedance 
is: 

Zi  =  ri  +  jsxi. 

Let  the  secondary  circuit  be  closed  by  an  external  resistance, 
r,  and  an  external  reactance,  and  denote  the  latter  by  x  at 
frequency,  /,  then  at  frequency,  s/,  or  slip,  s,  it  will  be  =  sx,  and 
thus: 

Z  =  r  +  jsx  =  external  secondary  impedance.1 

Let: 

EQ    =  primary  impressed  e.m.f.  per  circuit, 

E'    =  e.m.f.  consumed  by  primary  counter  e.m.f., 

EI    =  secondary  terminal  e.m.f., 

E'i  =  secondary  generated  e.m.f., 

e  =  e.m.f.  generated  per  turn  by  the  mutual  magnetic 

flux,  at  full  frequency,  /, 
/o   =  primary  current, 
I  QO  =  primary  exciting  current, 
/i    =  secondary  current. 

It  is  then: 

Secondary  generated  e.m.f.  : 


Total  secondary  impedance: 

Zl  +  Z  =  (n  +  r) 
hence,  secondary  current: 


Z,  +  Z       (n  +  r)  +  js  (xi  +  x) 

1  This  applies  to  the  case  where  the  secondary  contains  inductive  react- 
ance only;  or,  rather,  that  kind  of  reactance  which  is  proportional  to  the 
frequency.  In  a  condenser  the  reactance  is  inversely  proportional  to  the 
frequency,  in  a  synchronous  motor  under  circumstances  independent  of  the 
frequency.  Thus,  in  general,  we  have  to  set,  x  =  x'  +  x"  +  x'.",  where  x' 
is  that  part  of  the  reactance  which  is  proportional  to  the  frequency,  x"  that 
part  of  the  reactance  independent  of  the  frequency,  and  x'"  that  part  of  the 
reactance  which  is  inversely  proportional  to  the  frequency;  and  have  thus, 

x'" 
at  slip,  s,  or  frequency,  sf,  the  external  secondary  reactance,  sx'  +  x"  + 


180  ELECTRICAL  APPARATUS 

Secondary  terminal  voltage: 


j n  +  jsxi  \  sn±e  (r  +  jsx) 

~  (ri  +  r)  +  js  (xi  +  3)1  "  (rr  +  r)  +  j«  (*i  +  a;)' 

e.m.f.  consumed  by  primary  counter  e.m.f. 

77»/ 

jtv      =    Tlo^, 

hence,  primary  exciting  current: 

Joo  =  E'Y0  =  nQe  (g  -  jb). 

Component  of  primary  current  corresponding  to  secondary 
current,  J\\ 


n0se 


hence,  total  primary  current: 

/O    =   /OO  4~  J  8 

a  _J. flf  -  jb\ 

e\a2(rl  +  r)+js(xl  +  x)~}         s      J 

Primary  impressed  e.m.f. : 

EQ   =   E     ~\- 


We  get  thus,  as  the 

Equations  of  the  General  Alternating-current  Transformer,  of 
ratio  of  turns,  a;  and  ratio  of  frequencies,  s;  with  the  e.m.f. 
generated  per  turn  at  full  frequency,  6,  as  parameter,  the  values : 

Primary  impressed  e.m.f. : 

Secondary  terminal  voltage: 

TI  +  jsxi         _  |  _  r  +  jsx 


Primary  current: 

'_  _  [ _0  - 

la2  (n  +  r)  +  j 


r     ._  ^    •"• i     ¥ J 

J- o  —  sn$e  \ '  a  /      i     \    i    •   /       i    _\  n 


FREQUENCY  CONVERTER  181 

Secondary  current: 

sn\e 
= 


(ri  +  r)  +  J8  (xl  +  x) 

Therefrom,  we  get: 
Ratio  of  currents: 


J.  1         a    \. 

Ratio  of  e.m.fs.  : 

s 


a 


7*1  + 
1    — 


+  r)  +  js  (xi  +  x) 
Total  apparent  primary  impedance: 


1  +  T  to  -  J&)  [fri  +  r)  +  js  (x,  +  x 


where : 

x  =  x'  +'—  + 


in  the  general  secondary  circuit  as  discussed  in  footnote,  page  179. 
Substituting  in  these  equations : 


gives  the 

General  Equations  of  the  Stationary  Alternating-current  Transformer 
Substituting  in  the  equations  of  the  general  alternating-current 
transformer : 

Z  =  0, 
gives  the 

General  Equations  of  the  Induction  Motor 
Substituting: 


182  ELECTRICAL  APPARATUS 

and  separating  the  real  and  imaginary  quantities : 

f  r  s  ~\ 

E0  =  n0e  \    1  H j—j  (r0  (ri  +  r)  +  SXQ  (xi  +  x))  +  (r0#  -f  z0b) 

I  L  tt  £&  J 

Ps  ~~l  1 

-  j  [  -^  (sr0(xi  +  x)  -  x  (ri  +  r))  +  (ro&  -  &00r) J  | , 

H-  r  ,  ^i    •  rs(^i  +  z)  i 

^7- + SJ  -  4  -«w  -+ 


l  2 

Zk 

Neglecting  the  exciting  current,  or  rather  considering  it  as 
a  separate  and  independent  shunt  circuit  outside  of  the  trans- 
former, as  can  approximately  be  done,  and  assuming  the  primary 
impedance  reduced  to  the  secondary  circuit  as  equal  to  the 
secondary  impedance: 

Fo  =  0,  §  =  Zl. 

Substituting  this  in  the  equations  of  the  general  transformer 
we  get: 

• 

f  s 

Iri  (ri  +  r)  +  SXl  (Xi  +  x)l 

-•^[sTifa  +  x)  -^(n  +  r)]  P 
=          {[r  (n  +  r)  +  s*x  (x,  +  x)]  -  js  [rx,  -  xrj}, 


106.  The  true  power  is,  in  symbolic  representation 

Pr  v  T\  i 
:  I-"-*-    j 

denoting: 

-£  =  w 

gives : 

Secondary  output  of  the  transformer: 

PI  =  [Eili]1  =  (— H   r  =  srw; 


FREQUENCY  CONVERTER  183 

Internal  loss  in  secondary  circuit: 


Total  secondary  power: 


P1  +  P1i=    ^1B     (r  +  r,)=«w(r  +  r,); 

\  Zk    I 

Internal  loss  in  primary  circuit : 

Po1  =  *'oVo  =  s'oVitt2  =  (- 

\  Zk   > 

Total  electrical  output,  plus  loss: 

Total  electrical  input  of  primary: 

Po  =  [Eolo]1  =  s  (~l6]  2  (r  -f  ri  +  sri)  =  w  (r  +  n  +  sri) ; 

Hence,  mechanical  output  of  transformer: 

P  =  Po  -  P1  =  w  (1  -  s)  (r  +  ri); 
Ratio : 

mechanical  output  _P =  I  -  s  =  speed 

total  secondary  power       PI  +  Pi1  s  slip 

Thus, 

In  a  general  alternating  transformer  of  ratio  of  turns,  a,  and 
ratio  of  frequencies,  s,  neglecting  exciting  current,  it  is : 
Electrical  input  in  primary: 

s/ii262  (r  -\-  7*1  -|- 

Mechanical  output: 

s  (1  -  s)  nSe*  (r  + 


Electrical  output  of  secondary: 

_  _  s 
1  ~ 


Losses  in  transformer: 

p.+P^p,.        _lsW£*n__ 
n   "•"     '  "  (r,  +  r)2  +  s2  (xi  + 


184  ELECTRICAL  APPARATUS 

Of  these  quantities,  P1  and  PI  are  always  positive;  P0  and  P 
can  be  positive  or  negative,  according  to  the  value  of  s.  Thus 
the  apparatus  can  either  produce  mechanical  power,  acting  as 
a  motor,  or  consume  mechanical  power;  and  it  can  either  con- 
sume electrical  power  or  produce  electrical  power,  as  a  generator. 

107.  At: 

s  =  0,  synchronism,  P0  =  0,  P  =  0,  PI  =  0. 
At  0  <  s  <  1,  between  synchronism  and  standstill. 

Pi,  P  and  Po  are  positive;  that  is,  the  apparatus  consumes 
electrical  power,  P0,  in  the  primary,  and  produces  mechanical 
power,  P,  and  electrical  power,  PI  +  Pi1,  in  the  secondary,  which 
is  partly,  Pi1,  consumed  by  the  internal  secondary  resistance, 
partly,  PI,  available  at  the  secondary  terminals. 
In  this  case: 

P!  +  P!1  s 

P  "  1  -  s; 

that  is,  of  the  electrical  power  consumed  in  the  primary  circuit, 
Po,  a  part  Po1  is  consumed  by  the  internal  primary  resistance, 
the  remainder  transmitted  to  the  secondary,  and  divides  between 
electrical  power,  PI  +  Pi1,  and  mechanical  power,  P,  in  the 
proportion  of  the  slip,  or  drop  below  synchronism,  s,  to  the 
speed:  1  —  s. 

In  this  range,  the  apparatus  is  a  motor. 

At  s  >  1;  or  backward  driving,  P  <  0,  or  negative;  that  is, 
the  apparatus  requires  mechanical  power  for'  driving. 

Then: 

Po  -Po1  -Pi1  <Pi; 

that  is,  the  secondary  electrical  power  is  produced  partly  by 
the  primary  electrical  power,  partly  by  the  mechanical  power, 
and  the  apparatus  acts  simultaneously  as  transformer  and  as 
alternating-current  generator,  with  the  secondary  as  armature. 

The  ratio  of  mechanical  input  to  electrical  input  is  the  ratio 
of  speed  to  synchronism. 

In  this  case,  the  secondary  frequency  is  higher  than  the 
primary. 

At: 

s  <  0,  beyond  synchronism, 

P  <  0;  that  is,  the  apparatus  has  to  be  driven  by  mechanical 
power. 


FREQUENCY  CONVERTER          185 

Po  <  0;  that  is,  the  primary  circuit  produces  electrical  power 
from  the  mechanical  input. 

At: 

r  -f-  ri  +  sri  =  0,  or,  s  =  -  — J 

the  electrical  power  produced  in  the  primary  becomes  less  than 
required  to  cover  the  losses  of  power,  and  Po  becomes  positive 
again. 

We  have  thus: 


consumes    mechanical    and    primary    electric    power;    produces 
secondary  electric  power. 


consumes  mechanical,  and  produces  electrical  power  in  primary 
and  in  secondary  circuit. 

0  <  8  <  1 

consumes  primary  electric  power,  and  produces  mechanical  and 
secondary  electrical  power 

1  <  s 

consumes  mechanical  and  primary  electrical  power;  produces 
secondary  electrical  power. 

108.  As  an  example,  in  Fig.  62  are  plotted,  with  the  slip,  s,  as 
abscissae,  the  values  of: 

Secondary  electrical  output  as  Curve      I.; 
total  internal  loss  as  Curve    II.  ; 

mechanical  output  as  Curve  III.; 

primary  electrical  output      as  Curve  IV.; 


for  the  values : 


=  100.0;  r  =  0.4; 
=  0.1;  x  =  0.3; 
=  0.2; 


186 
hence: 


ELECTRICAL  APPARATUS 


Pi  = 


16,000  s\ 

1  +  s2  ' 


PoI  +  Pil  =  » 


1  +  s2  ' 
4000  s  (5  +  s) 

Po=     ~TT^~ 

p  =  20,000  s(l  -  s) 


GENERAL  ALTERNATE  CURRENT  TRANSFORMER 


2.22.01.81.61.41.21.0.8     .6    .4    .2      0     .2    .4     .6    .8    1.01.21.41.61.82.02.22.4 

FIG.  62. — Speed-power  curves  of  general  alternating- current  transformer. 

109.  Since  the  most  common  practical  application  of  the 
general  alternating-current  transformer  is  that  of  frequency 
converter,  that  is,  to  change  from  one  frequency  to  another, 
either  with  or  without  change  of  the  number  of  phases,  the 
following  characteristic  curves  of  this  apparatus  are  of  great 
interest : 

1.  The  regulation  curve;  that  is,  the  change  of  secondary 
terminal  voltage  as  function  of  the  load  at  constant  impressed 
primary  voltage. 


FREQUENCY  CONVERTER          187 

2.  The  compounding  curve;  th^t  is,  the  change  of  primary 
impressed  voltage  required  to  maintain  constant  secondary 
terminal  voltage. 

In  this  case  the  impressed  frequency  and  the  speed  are  con- 
stant, and  consequently  the  secondary  frequency  is  also  constant. 
Generally  the  frequency  converter  is  used  to  change  from  a  low 
frequency,  as  25  cycles,  to  a  higher  frequency,  as  60  or  62.5 
cycles,  and  is  then  driven  backward,  that  is,  against  its  torque, 
by  mechanical  power.  Mostly  a  synchronous  motor  is  em- 
ployed, connected  to  the  primary  mains,  which  by  overexcitation 
compensates  also  for  the  lagging  current  of  the  frequency 
converter. 

Let: 

Y  =  g  —  jb  =  primary  exciting  admittance  per  circuit  of 
the  frequency  converter. 

Zi  =  n  -f-  jxi  =  internal  self-inductive  impedance  per  sec- 
ondary circuit,  at  the  secondary  frequency. 

ZQ  =  rQ  +  jxQ  =  internal  self -inductive  impedance  per  primary 
circuit  at  the  primary  frequency. 

a    =  ratio  of  secondary  to  primary  turns  per  circuit. 

b  =  ratio  of  number  of  secondary  to  number  of  primary 
circuits. 

c     =  ratio  of  secondary  to  primary  frequencies. 

Let: 

e-=  generated  e.m.f.  per  secondary  circuit  at  secondary 
frequency. 

Z  =  r  +  jx  =  external  impedance  per  secondary  circuit  at 
secondary  frequency,  that  is  load  on  secondary  system,  where 
x  =  0  for  non-inductive  load. 

To  calculate  the  characteristics  of  the  frequency  converter, 
we  then  have: 
the  total  secondary  impedance: 

Z  +  Zi  =  (r +  n)  +j(x  +  si); 
the  secondary  current: 


Z_l_ 
_J— 

where : 


r  +  ri  x  + 

and  a2  = 


(r  +  rO2  +  (x  +  *i)2  (r  +  n)2  +  (x  +  x,}1 


188  ELECTRICAL  APPARATUS 

and  the  secondary  terminal  voltage  : 


=  e  (r  +  jo;)  (01  -  ja2)  =  e  (61  -  ^2)  ; 
where  : 

61  =  (rai  +  xaz)  and  62  =  (ra2  — 
primary  generated  e.m.f.  per  circuit: 

S'=^; 

ac 

primary  load  current  per  circuit: 

7l  =  06/1  =  abe  (0,1  -  jaz)  ; 
primary  exciting  current  per  circuit: 


« 
thus,  total  primary  current: 

/o  =  I1  +  /oo  =  e  (ci  -  jc2); 
where  : 

Ci  =  a6«i  +          and     c2  =  aba2  -\  --  ; 
ac  ac 

and  the  primary  terminal  voltage: 

E0  =  El 

where  : 

di  =  --  +  foCi  +  xQc2  and  d2  =  r0c2  —  xQCi; 
ac 

or  the  absolute  value  is  : 


substituting  this  value  of  e  in  the  preceding  equations,  gives, 
as  function  of  the  primary  impressed  e.m.f.,  eQ: 
secondary  current: 

7        eo  (PI  -  jo2)  j  laS  +  a22 


secondary  terminal  voltage: 


FREQUENCY  CONVERTER 


189 


primary  current: 


_  e°   Cl  ~ 

" 


J£±£t; 


primary  impressed  e.m.f . : 

#0     = 


-  jdt) 


£,.  , 

0. 

1 

£ 
13 

£?  SECONDA 
g  TERMINA 
VOLTAGE 

/ 

/ 

2500 

2400 
2300 

!• 

• 

.. 

'- 

• 

x 

.  

_12_ 

/ 

10 

x 

X 

9 

' 

/ 

' 

8 

X 

X 

7 

X 

g 

^ 

<*" 

^ 

REGULATION  CURVES 
PRIMARY,  6350  VOLTS  CONSTANT 

ii 

5 

25  CYCLES  THREE-PHASE 
SECONDARY,  62.5  CYCLES  QUARTER-PHASE 

4 

0 

1 

0 

i 

SECONDARY  CURRENT   PER   PHASE,   AMP. 

0                30                40                 50 

6 

0 

FIG.  63.  —  Regulation  curves  of  frequency  converter. 
secondary  output: 

P        firm      go2  (fli&i  +  a2fr2) 
Pl  =  [£/1/l!  dj  +  d,»  —  ; 

primary  electrical  input  : 


primary  apparent  input,  volt-amperes  : 


190 


ELECTRICAL  APPARATUS 


Substituting  thus  different  values  for  the  secondary  external 
impedance,  Z,  gives  the  regulation  curve  of  the  frequency 
converter. 

Such  a  curve,  taken  from  tests  of  a  200-kw.  frequency  converter 
changing  from  6300  volts,  25  cycles,  three-phase,  to  2500  volts, 
62.5  cycles,  quarter-phase,  is  given  in  Fig.  63. 


PRIMARY 

_—  — 

S^^^M 

_ 

^^—  —  ?r 



••.         • 

_     — 

/ 

VOLTS 

_6500_ 
6000 

AMP. 

_13_ 
-12- 
11_ 

S 

Y 

/ 

/ 

10 

/ 

/ 

9 

/ 

f 

8 

/ 

/ 

7 

/ 

S 

6 

' 

^ 

^^ 

COMPOUNDING  CURVES 
SECONDARY,  2500  VOLTS  CONSTAN1 
62.5  CYCLES  QUARTER-PHASE 
PRIMARY,  25  CYCLES  THREE-PHASE 

— 

5. 

4 

0 

1 

0 

2 

SE 

0 

3ONDAE 

s 

V   CUR 

o 

RENT  ( 

4 

>ER   PHASE,    AMP. 

0        I        50 

i 

FIG.  64. — Compounding  curve  of  frequency  converter. 

From  the  secondary  terminal  voltage: 
Ei  =  e  (bi-  j62), 
it  follows,  absolute: 


=  e 


e  = 


Substituting  these  values  in  the  above  equation  gives  the 
quantities  as  functions  of  the  secondary  terminal  voltage,  that 
is,  at  constant,'  e\,  or  the  compounding  curve. 

The  compounding  curve  of  the  frequency  converter  above 
mentioned  is  given  in  Fig.  64. 

110.  When  running  above  synchronism:  s  <  0,  the  general 
alternating-current  transformer  consumes  mechanical  power  and 


FREQUENCY  CONVERTER          191 

produces  electric  power  in  both  circuits,  primary  and  secondary, 
thus  can  not  be  called  a  frequency  converter,  and  the  distinc- 
tion between  primary  and  secondary  circuits  ceases,  but  both 
circuits  are  generator  circuits.  The  machine  then  is  a  two-fre- 
quency induction  generator.  As  the  electric  power  generated 
at  the  two  frequencies  is  proportional  to  the  frequencies,  this 
gives  a  limitation  to  the  usefulness  of  the  machine,  and  it  appears 
suitable  only  in  two  cases: 

(a)  If  s  =  —1,  both  frequencies  are  the  same,  and  stator 
and  rotor  circuits  can  be  connected  together,  in  parallel  or  in 
series,  giving  the  "  double  synchronous-induction  generator." 
Such  machines  have  been  proposed  for  steam-turbine  alternators 
of  small  and  moderate  sizes,  as  they  permit,  with  bipolar  con- 
struction, to  operate  at  twice  the  maximum  speed  available  for 
the  synchronous  machine,  which  is  1500  revolutions  for  25  cycles, 
and  3600  revolutions  for  60  cycles. 

(6)  If  s  is  very  small,  so  that  the  power  produced  in  the  low- 
frequency  circuit  is  very  small  and  may  be  absorbed  by  a  small 
"low-frequency  exciter." 

Further  discussion  of  both  of  these  types  is  given  in  the 
Chapter  XIII  on  the  "  Synchronous  Induction  Generator." 

111.  The  use  of  the  general  alternating-current  transformer  as 
frequency  converter  is  always  accompanied  by  the  production 
of  mechanical  power  when  lowering,  and  by  the  consumption 
of  mechanical  power  when  raising  the  frequency.  Thus  a  second 
machine,  either  induction  or  synchronous,  would  be  placed  on  the 
frequency  converter-  shaft  to  supply  the  mechanical  power  as 
motor  when  raising  the  frequency,  or  absorb  the  power  as 
generator,  when  lowering  the  frequency.  This  machine  may  be 
of  either  of  the  two  frequencies,  but  would  naturally,  for  eco- 
nomical reasons,  be  built  for  the  supply  frequency,  when  motor, 
and  for  the  generated  or  secondary  frequency,  when  generator. 

Such  a  couple  of  frequency  converter  and  driving  motor  and 
auxiliary  generator  has  over  a  motor-generator  set  the  advan- 
tage, that  it  requires  a  total  machine  capacity  only  equal  to  the 
output,  while  with  a  motor-generator  set  the  total  machine 
capacity  equals  twice  the  output.  It  has,  however,  the  dis- 
advantage not  to  be  as  standard  as  the  motor  and  the  generator. 

If  a  synchronous  machine  is  used,  the  frequency  is  constant; 
if  an  induction  machine  is  used,  there  is  a  slip,  increasing  with 
the  load,  that  is,  the  ratio  of  the  two  frequencies  slightly  varies 


192  ELECTRICAL  APPARATUS 

with  the  load,  so  that  the  latter  arrangement  is  less  suitable  when 
tying  together  two  systems  of  constant  frequencies. 

112.  Frequency  converters  may  be  used: 

(a)  For  producing  a  moderate  amount  of  power  of  a  higher  or 
a  lower  frequency,  from  a  large  alternating-current  system. 

(6)  For  tying  together  two  alternating-current  systems  of 
different  frequencies,  and  interchange  power  between  them,  so 
that  either  acts  as  reserve  to  the  other.  In  this  case,  electrical 
power  transfer  may  be  either  way. 

(c)  For  local  frequency  reduction  for  commutating  machines, 
by  having  the  general  alternating-current  transformer  lower  the 
frequency,  for  instance  from  60  to  30  cycles,  and  take  up  the 
lower  frequency,  as  well  as  the  mechanical  power  in  a  commu- 
tating machine   on  the    frequency    converter    shaft.     Such    a 
combination  has  been  called  a  "Motor  Converter." 

Thus,  instead  of  a  60-cycle  synchronous  converter,  such  a 
60/30-cycle  motor  converter  would  offer  the  advantage  of  the 
lower  frequency  of  30  cycles  in  the  commutating  machine.  The 
commutating  machine  then  would  receive  half  its  input  electric- 
ally, as  synchronous  converter,  half  mechanically,  as  direct- 
current  generator,  and  thus  would  be  half  converter  and  half 
generator;  the  induction  machine  on  the  same  shaft  would  change 
half  of  its  60-cycle  power  input  into  mechanical  power,  half  into 
30-cycle  electric  power. 

Such  motor  converter  is  smaller  and  more  efficient  than  a 
motor-generator  set,  but  larger  and  less  efficient  than  a  syn- 
chronous converter. 

Where  phase  control  of  the  direct-current  voltage  is  desired, 
the  motor  converter  as- a  rule  does  not  require  reactors,  as  the 
induction  machine  has  sufficient  internal  reactance. 

(d)  For  supplying  low  frequency  to  a  second  machine  on  the 
same  shaft,  for  speed  control,  as  "  concatenated  motor  couple." 
That  is,  two  induction  motors   on  the   same   shaft,  operating 
in   parallel,    give    full    speed,    and    half    speed    is    produced, 
at   full   efficiency,    by   concatenating    the    two    induction  ma- 
chines, that  is,  using  the  one  as  frequency  converter  for  feeding 
the  other. 

By  using  two  machines  of  different  number  of  poles,  p:  and 
p2,  on  the  same  shaft,  four  different  speeds  can  be  secured,  corre- 
sponding respectively  to  the  number  of  poles:  p^  +  Pz,  PI,  Pi, 
pi  —  p%.  That  is,  concatenation  of  both  machines,  operation 


FREQUENCY  CONVERTER          193 

of  one  machine  only,  either  the  one  or  the  other,  and  differential 
concatenation. 

Further  discussion  hereof  see  under  "  Concatenation." 
In  some  forms  of  secondary  excitation  of  induction  machines, 
as  by  low-frequency  synchronous  or  commutating  machine  in 
the  secondary,  the  induction  machine  may  also  be  considered 
as  frequency  converter.  Regarding  hereto  see  "  Induction 
Motors  with  Secondary  Excitation." 


L3 


CHAPTER  XIII 
SYNCHRONOUS  INDUCTION  GENERATOR 

113.  If  an  induction  machine  is  driven  above  synchronism, 
the  power  component  of  the  primary  current  reverses,  that  is, 
energy  flows  outward,  and  the  machine  becomes  an  induction 
generator.  The  component  of  current  required  for  magnetiza- 
tion remains,  however,  the  same;  that  is,  the  induction  generator 
requires  the  supply  of  a  reactive  current  for  excitation,  just  as 
the  induction  motor,  and  so  must  be  connected  to  some  apparatus 
which  gives  a  lagging,  or,  what  is  the  same,  consumes  a  leading 
current. 

The  frequency  of  the  e.m.f.  generated  by  the  induction  gen- 
erator, /,  is  lower  than  the  frequency  of  rotation  or  speed,  /0, 
by  the  frequency,  /],  of  the  secondary  currents.  Or,  inversely, 
the  frequency,  /i,  of  the  secondary  circuit  is  the  frequency  of 
slip — that  is,  the  frequency  with  which  the  speed  of  mechanical 
rotation  slips  behind  the  speed  of  the  rotating  field,  in  the  induc- 
tion motor,  or  the  speed  of  the  rotating  field  slips  behind  the 
speed  of  mechanical  rotation,  in  the  induction  generator. 

As  in  every  transformer,  so  in  the  induction  machine,  the 
secondary  current  must  have  the  same  ampere-turns  as  the 
primary  current  less  the  exciting  current,  that  is,  the  secondary 
current  is  approximately  proportional  to  the  primary  current, 
or  to  the  load  of  the  induction  generator. 

In  an  induction  generator  with  short-circuited  secondary, 
the  secondary  currents  are  proportional,  approximately,  to  the 
e.m.f.  generated  in  the  secondary  circuit,  and  this  e.m.f.  is  pro- 
portional to  the  frequency  of  the  secondary  circuit,  that  is, 
the  slip  of  frequency  behind  speed.  It  so  follows  that  the  slip 
of  frequency  in  the  induction  generator  with  short-circuited 
secondary  is  approximately  proportional  to  the  load,  that  is, 
such  an  induction  generator  does  not  produce  constant  syn- 
chronous frequency,  but  a  frequency  which  decreases  slightly 
with  increasing  load,  just  as  the  speed  of  the  induction  motor 
decreases  slightly  with  increase  of  load. 

Induction  generator  and  induction  motor  so  have  also  been 

194 


SYNCHRONOUS  INDUCTION  GENERATOR       195 

called  asynchronous  generator  and  asynchronous  motor,  but 
these  names  are  wrong,  since  the  induction  machine  is  not 
independent  of  the  frequency,  but  depends  upon  it  just  as  much 
as  a  synchronous  machine — the  difference  being,  that  the 
synchronous  machine  runs  exactly  in  synchronism,  while  the 
induction  machine  approaches  synchronism.  The  real  asyn- 
chronous machine  is  the  commutating  machine. 

114.  Since  the  slip  of  frequency  with  increasing  load  on  the 
induction  generator  with  short-circuited  secondary  is  due  to 
the  increase  of  secondary  frequency  required  to  produce  the 
secondary  e.m.f.  and  therewith  the  secondary  currents,  it  follows: 
if  these  secondary  currents  are  produced  by  impressing  an  e.m.f. 
of  constant  frequency,  fi,  upon  the  secondary  circuit,  the  primary 
frequency,  /,  does  not  change  with  the  load,  but  remains  con- 
stant and  equal  to  /  =  /0  —  /i.  The  machine  then  is  a  syn- 
chronous-induction machine — that  is,  a  machine  in  which  the 
speed  and  frequency  are  rigid  with  regard  to  each  other,  just  as 
in  the  synchronous  machine,  except  that  in  the  synchronous- 
induction  machine,  speed  and  frequency  have  a  constant  dif- 
ference, while  in  the  synchronous  machine  this  difference  is  zero, 
that  is,  the  speed  equals  the  frequency. 

By  thus  connecting  the  secondary  of  the  induction  machine 
with  a  source  of  constant  low-frequency,  /i,  as  a  synchronous 
machine,  or  a  commutating  machine  with  low-frequency  field 
excitation,  the  primary  of  the  induction  machine  at  constant 
speed,  /o,  generates  electric  power  at  constant  frequency,  /, 
independent  of  the  load.  If  the  secondary  /]  =  0,  that  is,  a 
continuous  current  is  supplied  to  the  secondary  circuit,  the 
primary  frequency  is  the  frequency  of  rotation  and  the  machine 
an  ordinary  synchronous  machine.  The  synchronous  machine  so 
appears  as  a  special  case  of  the  synchronous-induction  machine 
and  corresponds  to  /]  =  0. 

In  the  synchronous-induction  generator,  or  induction  machine 
with  an  e.m.f.  of  constant  low  frequency,  /i,  impressed  upon  the 
secondary  circuit,  by  a  synchronous  machine,  etc.,  with  increas- 
ing load,  the  primary  and  so  the  secondary  currents  change,  and 
the  synchronous  machine  so  receives  more  power  as  synchronous 
motor,  if  the  rotating  field  produced  in  the  secondary  circuit 
revolves  in  the  same  direction  as  the  mechanical  rotation — 
that  is,  if  the  machine  is  driven  above  synchronism  of  the 
e.m.f.  impressed  upon  the  secondary  circuit — or  the  synchronous 


196  ELECTRICAL  APPARATUS 

machine  generates  more  power  as  alternator,  if  the  direction  of 
rotation  of  the  secondary  revolving  field  is  in  opposition  to  the 
speed.  In  the  former  case,  the  primary  frequency  equals  speed 
minus  secondary  impressed  frequency :  /  =  /0  —  /i ;  in  the  latter 
case,  the  primary  frequency  equals  the  sum  of  speed  and  sec- 
ondary impressed  frequency:  f  =  fo  + /i,  and  the  machine  is  a 
frequency  converter  or  general  alternating-current  transformer, 
with  the  frequency,  /i,  as  primary,  and  the  frequency,  /,  as 
secondary,  transforming  up  in  frequency  to  a  frequency,  /, 
which  is  very  high  compared  with  the  impressed  frequency, 
so  that  the  mechanical  power  input  into  the  frequency  con- 
verter is  very  large  compared  with  the  electrical  power  input. 

The  synchronous-induction  generator,  that  is,  induction  gen- 
erator in  which  the  secondary  frequency  or  frequency  of  slip  is 
fixed  by  an  impressed  frequency,  so  can  also  be  considered  as  a 
frequency  converter  or  general  alternating-current  transformer. 

115.  To  transform  from  a  frequency,  /i,  to  a  frequency;  /2,  the 
frequency,  /i,  is  impressed  upon  the  primary  of  an  induction 
machine,  and  the  secondary  driven  at  such  a  speed,  or  fre- 
quency of  rotation,  /0,  that  the  difference  between  primary 
impressed  frequency,  /i,  and  frequency  of  rotation,  /0,  that  is, 
the  frequency  of  slip,  is  the  desired  secondary  frequency,  /2. 

There  are  two  speeds,  /0,  which  fulfill  this  condition:  one 
below  synchronism :  /0  =  f\  —  /2,  and  one  above  synchronism : 
J0  =  fl  -f  /2.  That  is,  the  secondary  frequency  becomes  /2, 
if  the  secondary  runs  slower  than  the  primary  revolving  field 
of  frequency,  /i,  or  if  the  secondary  runs  faster  than  the  primary 
field,  by  the  slip,  /2. 

In  the  former  case,  the  speed  is  below  synchronism,  that  is, 
the  machine  generates  electric  power  at  the  frequency,  /2,  in  the 
secondary,  and  consumes  electric  power  at  the  frequency,  /], 
in  the  primary.  If  /2  <  /i,  the  speed  /0  =  /i  —  /2  is  between 
standstill  and  synchronism,  and  the  machine,  in  addition  to 
electric  power,  generates  mechanical  power,  as  induction  motor, 
and  as  has  been  seen  in  the  chapter  on  the  "  General  Alternating- 
current  Transformer,"  it  is,  approximately: 

Electric  power  input  -r-  electric  power  output  -f-  mechanical 
power  output  =  f\  -f-  /2  -r-  /0. 

If  /2  >'/i,  that  is,  the  frequency  converter  increases  the  fre- 
quency, the  rotation  must  be  in  backward  direction,  against  the 
rotating  field,  so  as  to  give  a  slip,  /2,  greater  than  the  impressed 


SYNCHRONOUS  INDUCTION  GENERATOR       197 

frequency,  /i,  and  the  speed  is  /o  =  /2  —  fi-  In  this  case,  the 
machine  consumes  mechanical  power,  since  it  is  driven  against 
the  torque  given  by  it  as  induction  motor,  and  we  have: 

Electric  power  input  -s-.  mechanical  power  input  -f-  electric 
power  output  =  fi  -j-  /0  -f-  /2. 

That  is,  the  three  powers,  primary  electric,  secondary  electric, 
and  mechanical,  are  proportional  to  their  respective  frequencies. 

As  stated,  the  secondary  frequency,  /2,  is  also  produced  by 
driving  the  machine  above  synchronism,  /],  that  is,  with  a 
negative  slip,  /2,  or  at  a  speed,  /o  =  /i  +  /2-  In  this  case,  the 
machine  is  induction  generator,  that  is,  the  primary  circuit 
generates  electric  power  at  frequency  /i,  the  secondary  circuit 
generates  electric  power  at  frequency  /2,  and  the  machine  con- 
sumes mechanical  power,  and  the  three  powers  again  are  propor- 
tional to  their  respective  frequencies: 

Primary  electric  output  -r-  secondary  electric  output  -r- 
mechanical  input  =  J\  -i-  /2  -f-  /o. 

Since  in  this  case  of  oversynchronous  rotation,  both  electric 
circuits  of  the  machine  generate,  it  can  not  be  called  a  frequency 
converter,  but  is  an  electric  generator,  converting  mechanical 
power  into  electric  power  at  two  different  frequencies,  /i  and 
fz,  and  so  is  called  a  synchronous-induction  machine,  since 
the  sum  of  the  two  frequencies  generated  by  it  equals  the  fre- 
quency of  rotation  or  speed — that  is,  the  machine  revolves  in 
synchronism  with  the  sum  of  the  two  frequencies  generated 
by  it. 

It  is  obvious  that  like  all  induction  machines,  this  synchro- 
nous-induction generator  requires  a  reactive  lagging  current  for 
excitation,  which  has  to  be  supplied  to  it  by  some  outside  source, 
as  a  synchronous  machine,  etc. 

That  is,  an  induction  machine  driven  at  speed,  /o,  when  sup- 
plied with  reactive  exciting  current  of  the  proper  frequency, 
generates  electric  power  in  the  stator  as  well  as  in  the  rotor,  at 
the  two  respective  frequencies,  /i  and  /2,  which  are  such  that  their 
sum  is  in  synchronism  with  the  speed,  that  is : 

/.  +  /.  =  /o; 

otherwise  the  frequencies,  /i  and  /2,  are  entirely  independent. 
That  is,  connecting  the  stator  to  a  circuit  of  frequency,  /i,  the 
rotor  generates  frequency,  /2  =  /o  —  /i,  or  connecting  the  rotor  to 


198  ELECTRICAL  APPARATUS 

a   circuit   of  frequency,  /2,   the   stator  generates   a  frequency 

7l    =/0-/2. 

116.  The  power  generated  in  the  stator,  PI,  and  the  power 
generated  in  the  rotor,  P2,  are  proportional  to  their  respective 
frequencies : 

PI  :  P2  '  PQ  =  fi  :  /2  :  fo, 

where  P0  is  the  mechanical  input  (approximately,  that  is,  neg- 
lecting losses). 

As  seen  here  the  difference  between  the  two  circuits,  stator 
and  rotor,  disappears — that  is,  either  can  be  primary  or  sec- 
ondary, that  is,  the  reactive  lagging  current  required  for  excita- 
tion can  be  supplied  to  the  stator  circuit  at  frequency,  /i,  or  to 
the  rotor  circuit  at  frequency, /2,  or  a  part  to  the  stator  and  a  part 
to  the  rotor  circuit.  Since  this  exciting  current  is  reactive  or 
wattless,  it  can  be  derived  from  a  synchronous  motor  or  con- 
verter, as  well  as  from  a  synchronous  generator,  or  an  alter- 
nating commutating  machine. 

As  the  voltage  required  by  the  exciting  current  is  proportional 
to  the  frequency,  it  also  follows  that  the  reactive  power  input  or 
the  volt-amperes  excitation,  is  proportional  to  the  frequency 
of  the  exciting  circuit.  Hence,  using  the  low-frequency  circuit 
for  excitation,  the  exciting  volt-amperes  are  small. 

Such  a  synchronous-induction  generator  therefore  is  a  two- 
frequency  generator,  producing  electric  power  simultaneously 
at  two  frequencies,  and  in  amounts  proportional  to  these  fre- 
quencies. For  instance,  driven  at  85  cycles,  it  can  connect  with 
the  stator  to  a  25-cycle  system,  and  with  the  rotor  to  a  60-cycle 
system,  and  feed  into  both  systems  power  in  the  proportion  of 
25  -T-  60,  as  is  obvious  from  the  equations  of  the  general  alter- 
nating-current transformer  in  the  preceding  chapter 

117.  Since  the   amounts   of  electric   power  at  the  two  fre- 
quencies are  always  proportional  to  each  other,  such  a  machine 
is  hardly  of  much  value  for  feeding  into  two  different  systems, 
but  of  importance  are  only  the  cases  where  the  two  frequencies 
generated  by  the  machine  can  be  reduced  to  one. 

This  is  the  case: 

1.  If  the  two  frequencies  are  the  same:  /i  =  /2  =  ??•     In  this 

case,  stator  and  rotor  can  be  connected  together,  in  parallel 
or  in  series,  and  the  induction  machine  then  generates  electric 
power  at  half  the  frequency  of  its  speed,  that  is,  runs  at  double 


SYNCHRONOUS  INDUCTION  GENERATOR       199 

synchronism  of  its  generated  frequency.  Such  a  "double  syn- 
chronous alternator"  so  consists  of  an  induction  machine,  in 
which  the  stator  and  the  rotor  are  connected  with  each  other  in 
parallel  or  in  series,  supplied  with  the  reactive  exciting  current 
by  a  synchronous  machine — for  instance,  by  using  synchronous 
converters  with  overexcited  field  as  load — and  driven  at.  a  speed 
equal  to  twice  the  frequency  required.  This  type  of  machine 
may  be  useful  for  prime  movers  of  very  high  speeds,  such  as 
steam  turbines,  as  it  permits  a  speed  equal  to  twice  that  of  the 
bipolar  synchronous  machine  (3000  revolutions  at  25,  and  7200 
revolutions  at  60  cycles). 

2.  If  of  the  two  frequencies,  one  is  chosen  so  low  that  the 
amount  of  power  generated  at  this  frequency  is  very  small,  and 
can  be  taken  up  by  a  synchronous  machine  or  other  low-fre- 
quency machine,  the  latter  then  may  also  be  called  an  exciter. 
For  instance,  connecting  the  rotor  of  an  induction  machine  to  a 
synchronous  motor  of  /2  =  4  cycles,  and  driving  it  at  a  speed 
of  /o  =  64  cycles,  generates  in  the  stator  an  e.m.f.  at  f\  =  60 
cycles,  and  the  amount  of  power  generated  at  60  cycles  is  6%  = 
15  times  the  power  generated  by  4  cycles.  The  machine  then 
is  an  induction  generator  driven  at  15  times  its  synchronous 
speed.  Where  the  power  at  frequency,  /2,  is  very  small,  it  would 
be  no  serious  objection  if  this  power  were  not  generated,  but  con- 
sumed. That  is,  by  impressing  /2  =  4  cycles  upon  the  rotor, 
and  driving  it  at  /0  =  56  cycles,  in  opposite  direction  to  the  rotat- 
ing field  produced  in  it  by  the  impressed  frequency  of  4  cycles, 
the  stator  also  generates  an  e.m.f.  at  f\  —  60  cycles.  In  this 
case,  electric  power  has  to  be  put  into  the  machine  by  a  generator 
at  /2  =  4  cycles,  and  mechanical  power  at  a  speed  of  /0  =  56 
cycles,  and  electric  power  is  produced  as  output  at/i  =  60  cycles. 
The  machine  thus  operated  is  an  ordinary  frequency  converter, 
which  transforms  from  a  very  low  frequency,  /2  =  4  cycles,  to 
frequency  fi  =  60  cycles  or  15  times  the  impressed  frequency, 
and  the  electric  power  input  so  is  only  one-fifteenth  of  the  electric 
power  output,  the  other  fourteen-fifteenths  are  given  by  the 
mechanical  power  input,  and  the  generator  supplying  the  im- 
pressed frequency,  /2  =  4  cycles,  accordingly  is  so  small  that  it 
can  be  considered  as  an  exciter. 

118.  3.  If  the  rotor  of  frequency,  /2,  driven  at  speed,  /o,  is 
connected  to  the  external  circuit  through  a  commutator,  the 
effective  frequency  supplied  by  the  commutator  brushes  to  the 


200  ELECTRICAL  APPARATUS 

external  circuit  is/0  —  /2;  hence  equals  /i,  or  the  stator  frequency. 
Stator  and  rotor  so  give  the  same  effective  frequency,  /i,  and 
irrespective  of  the  frequency,  /2  generated  in  the  rotor,  and  the 
frequencies,  /i  and  /2,  accordingly  become  indefinite,  that  is, 
/i  may  be  any  frequency.  /2  then  becomes  /0  —  /],  but  by  the 
commutator  is  transformed  to  the  same  frequency,  /i.  If  the 
stator  and  rotor  were  used  on  entirely  independent  electric 
circuits,  the  frequency  would  remain  indeterminate.  As  soon, 
however,  as  stator  and  rotor  are  connected  together,  a  relation 
appears  due  to  the  transformer  law,  that  the  secondary  ampere- 
turns  must  equal  the  primary  ampere-turns  (when  neglecting 
the  exciting  ampere-turns)  .  This  makes  the  frequency  dependent 
upon  the  number  of  turns  of  stator  and  rotor  circuit. 

Assuming  the  rotor  circuit  is  connected  in  multiple  with  the 
stator  circuit  —  as  it  always  can  be,  since  by  the  commutator 
brushes  it  has  been  brought  to  the  same  frequency.  The  rotor 
e.m.f.  then  must  be  equal  to  the  stator  e.m.f.  The  e.m.f.,  how- 
ever, is  proportional  to  the  frequency  times  number  of  turns, 
and  it  is  therefore: 

^2/2    =    Wi/i, 

where:  ,        n\  =  number  of  effective  stator  turns, 

ri2  =  number  of  effective  rotor  turns,  and  f\ 

and  /2  are  the  respective  frequencies. 
Herefrom  follows: 


that  is,  the  frequencies  are  inversely  proportional  to  the  number 
of  effective  turns  in  stator  and  in  rotor. 

Or,  since  /o  =  /i  +  h  is  the  frequency  of  rotation  : 

/i  ^  /o  =  ^2  -5-  wi  +  nz, 


That  is,  the  frequency,  /i,  generated  by  the  synchronous- 
induction  machine  with  commutator,  is  the  frequency  of  rotation, 
/o,  times  the  ratio  of  rotor  turns,  n2,  to  total  turns,  ni  -f-  n^. 

Thus,  it  can  be  made  anything  by  properly  choosing  the 
number  of  turns  in  the  rotor  and  in  the  stator,  or,  what  amounts 
to  the  same,  interposing  between  rotor  and  stator  a  transformer 
of  the  proper  ratio  of  transformation. 


SYNCHRONOUS  INDUCTION  GENERATOR       201 

The  powers  generated  by  the  stator  and  by  the  rotor,  how- 
ever, are  proportional  to  their  respective  frequencies,  and  so  are 
inversely  proportional  to  their  respective  turns. 

PI  -r-  PZ  =  fi  -r-  /2  =  HZ  -5-  n\\ 

if  n\  and  n2,  and  therewith  the  two  frequencies,  are  very  different, 
the  two  powers,  Pj  and  P2,  are  very  different,  that  is,  one  of  the 
elements  generates  very  much  less  power  than  the  other,  and 
since  both  elements,  stator  and  rotor,  have  the  same  active 
surface,  and  so  can  generate  approximately  the  same  power,  the 
machine  is  less  economical. 

That  is,  the  commutator  permits  the  generation  of  any  de- 
sired frequency,  /i,  but  with  best  economy  only  if  /i  =  ^,  or 

half-synchronous  frequency,  and  the  greater  the  deviation  from 
this  frequency,  the  less  is  the  economy.  If  one  of  the  fre- 
quencies is  very  small,  that  is,  /i  is  either  nearly  equal  to  syn- 
chronism, /0,  or  very  low,  the  low-frequency  structure  generates 
very  little  power. 

By  shifting  the  commutator  brushes,  a  component  of  the  rotor 
current  can  be  made  to  magnetize  and  the  machine  becomes  a 
self-exciting,  alternating-current  generator. 

The  use  of  a  commutator  on  alternating-current  machines  is 
in  general  undesirable,  as  it  imposes  limitations  on  the  design, 
for  the  purpose  of  eliminating  destructive  sparking,  as  discussed 
in  the  chapter  on  " Alternating-Current  Commutating  Machines." 

The  synchronous-induction  machines  have  not  yet  reached  a 
sufficient  importance  to  require  a  detailed  investigation,  so  only 
two  examples  may  be  considered. 

119.  1.  Double  Synchronous  Alternator. 

Assume  the  stator  and  rotor  of  an  induction  machine  to  be 
wound  for  the  same  number  of  effective  turns  and  phases,  and 
connected  in  multiple  or  in  series  with  each  other,  or,  if  wound 
for  different  number  of  turns,  connected  through  transformers 
of  such  ratios  as  to  give  the  same  effective  turns  when  reduced 
the  same  circuit  by  the  transformer  ratio  of  turns. 

Let: 

YI  =  g   —  jb    =  exciting  admittance  of  the  stator, 

Zi  =  7*1  -+-  jxi  =  self-inductive  impedance  of  the  stator, 

Zz  =  7*2  +  jxz  =  self-inductive  impedance  of  the  rotor, 


202  ELECTRICAL  APPARATUS 

and: 

e  =  e.m.f.  generated  in  the  stator  by  the  mutual  inductive 
magnetic  field,  that  is,  by  the  magnetic  flux  corresponding  to 
the  exciting  admittance,  F] ; 
and: 

/  =  total  current,  or  current  supplied  to  the  external  circuit, 
I\  =  stator  current, 
1 2  =  rotor  current. 

With  series  connection  of  stator  and  rotor: 

/    =    /I    =    /2, 

with  parallel  connection  of  stator  and  rotor: 

/    =    /I    +  /2- 

Using  the  equations  of  the  general  induction  machine,  the 
slip  of  the  secondary  circuit  or  rotor  is : 

*  =  -i; 

the  exciting  admittance  of  the  rotor  is: 

Yz  =  g  -  jsb  =  g  +  jb, 
and  the  rotor  generated  e.m.f. : 

E'z  =  se  =  —e; 

that  is,  the  rotor  must  be  connected  to  the  stator  in  the  opposite 
direction  to  that  in  which  it  would  be  connected  at  standstill, 
or  in  a  stationary  transformer. 

That  is,  magnetically,  the  power  components  of  stator  and 
rotor  current  neutralize  each  other.  Not  so,  however,  the 
reactive  components,  since  the  reactive  component  of  the  rotor 
current : 

/2    =    i\   +  ji»tt 

in  its  reaction  on  the  stator  is  reversed,  by  the  reversed  direction 
of  relative  rotation,  or  the  slip,  s  =  —  1,  and  the  effect  of  the 
rotor  current,  /2,  on  the  stator  circuit  accordingly  corresponds 
to: 

/'2    =   i>9   -  ji"2; 

hence,  the  total  magnetic  effect  is: 

fir-. IV.-  (i\-i'*}  +  j(*"i  +  *"2); 


SYNCHRONOUS  INDUCTION  GENERATOR       203 

and  since  the  total  effect  must  be  the  exciting  current: 

/o  =  i'Q  +  j"o, 
it  follows  that: 

i'i  -  i'z  =  i'o  and  i'\  +  i"2  =  i"0. 

Hence,  the  stator  power  current  and  rotor  power  current, 
i'i  and  i'%,  are  equal  to  each  other  (when  neglecting  the  small 
hysteresis  power  current).  The  synchronous  exgiter  of  the 
machine  must  supply  in  addition  to  the  magnetizing  current, 
the  total  reactive  current  of  the  load.  Or  in  other  words,  such 
a  machine  requires  a  synchronous  exciter  of  a  volt-ampere 
capacity  equal  to  the  volt-ampere  excitation  plus  the  reactive 
volt-amperes  of  the  load,  that  is,  with  an  inductive  load,  a  large 
exciter  machine.  In  this  respect,  the  double-synchronous 
generator  is  analogous  to  the  induction  generator,  and  is  there- 
fore suited  mainly  to  a  load  with  leading  current,  as  over- 
excited converters  and  synchronous  motors,  in  which  the  reactive 
component  of  the  load  is  negative  and  so  compensates  for  the 
reactive  component  of  excitation,  and  thereby  reduces  the  size 
of  the  exciter. 

This  means  that  the  double-synchronous  alternator  has  zero 
armature  reaction  for  non-inductive  load,  but  a  demagnetizing 
armature  reaction  for  inductive,  a  magnetizing  armature  reac- 
tion for  anti-inductive  load,  and  the  excitation,  by  alternating- 
reactive  current,  so  has  to  be  varied  with  the  character  of  the 
load,  in  general  in  a  far  higher  degree  than  with  the  synchronous 
alternator. 

120.  2.  Synchronous-induction  Generator  with  Low-frequency 
Excitation. 

Here  two  cases  exist: 

(a)  If  the  magnetic  field  of  excitation  revolves  in  opposite 
direction  to  the  mechanical  rotation. 

(b)  If  it  revolves  in  the  same  direction. 

In  the  first  case  (a)  the  exciter  is  a  low-frequency  generator 
and  the  machine  a  frequency  converter,  calculated  by  the  same 
equations. 

Its  voltage  regulation  is  essentially  that  of  a  synchronous 
alternator:  with  increasing  load,  at  constant  voltage  impressed 
upon  the  rotor  or  exciter  circuit,  the  voltage  drops  moderately 
at  non-inductive  load,  greatly  at  inductive  load,  and  rises  at 


204  ELECTRICAL  APPARATUS 

anti-inductive  load.  To  maintain  constant  terminal  voltage, 
the  excitation  has  to  be  changed  with  a  change  of  load  and 
character  of  load.  With  a  low-frequency  synchronous  machine 
as  exciter,  this  is  done  by  varying  the  field  excitation  of  the 
exciter. 

At  constant  field  excitation  of  the  synchronous  exciter,  the 
regulation  is  that  due  to  the  impedance  between  the  nominal 
generated  e.m.f.  of  the  exciter,  and  the  terminal  voltage  of  the 
stator — th%t  is,  corresponds  to: 

Z  =  Z0  +  Z2  +  Z,. 

Here  Z0  =  synchronous  impedance  of  the  exciter,  reduced  to  full 

frequency,  /i, 
Z2  =  self-inductive  impedance  of  the  rotor,  reduced  to  full 

frequency,  /i, 
Zi  =  self-inductive  impedance  of  the  stator.  \ 

If  then  EQ  =  nominal  generated  e.m.f.  of  the  exciter  generator, 
that  is,  corresponding  to  the  field  excitation,  and, 
Ii  =  i  —  ji\  =  stator    current    or    output    current,    the    stator 
terminal  voltage  is : 

Ei  =  E0  +  Z7,,  or,  Eo  =  E  +  (r  +  jx)  (i  -  jii) ; 
and,  choosing  EI  =  d  as  real  axis,  and  expanding: 

EQ  =  (ei  +  ri  +  xi})  +  j  (xi  —  rii), 
and  the  absolute  value: 

e02  =  (ei  +  ri  -f  xii)2  +  (xi  —  n'i)2, 
€l  =  Veo2  -  (xi  -  n'i)2  -  (ri  +  xii). 

121.  As  an  example  is  shown,  in  Fig.  65,  in  dotted  lines,  with 
the  total  current,  /  =  \/i2  +  i^t  as  abscissae,  the  voltage  regu- 
lation of  such  a  machine,  or.  the  terminal  voltage,  e\,  with  a 
four-cycle  synchronous  generator  as  exciter  of  the  60-cycle 
synchronous-induction  generator,  driven  as  frequency  converter 
at  56  cycles. 

1.  For  non-inductive  load,  or  /i  =  i.         (Curve  I.) 

2.  For  inductive  load  of  80  per  cent,  power-factor,  or  /i  = 
/  (0.8  -  0.6  j).  (Curve  II.) 

3.  For  anti-inductive  load  of  80  per  cent,  power-factor,  or 
II  =  I  (0.8  +  0.6  j).  (Curve  III.) 


SYNCHRONOUS  INDUCTION  GENERATOR       205 

For  the  constants: 


eQ   =  2000  volts, 
Zi  =  0.1  +  0.3  j, 


Z2  =  I  +  0.5  j, 
ZQ  =  0.5  +  0.5  j; 


hence: 
Then: 


=  1.6  +  1.3  j. 


6l  =  \/4  X  106  -  (1.3  i  -  1.6  ti)2  -  (1.6  *  -f  1.3  ii) ; 
hence,  for  non-inductive  load,  i\  =  0: 

ej  =  V4  X  106  -  1.69  z2  -  1.6  i; 


I 

2800 


2200 
2000 


1600 
1400 


1200 
1000 


bOO 


•200 


III 


III 


II 


500    600    700    800    900   1000 

AMPERES 


0    100    200    300    4 

FIG.  65. — Synchronous  induction  generator  regulation  curves. 

for  inductive  load  of  80  per  cent,  power-factor  ii  =  0.6  /,  i  =  0.8  7: 

ci  =  A/4  X  106  -  0.0064  P  -  2.067; 

and  for  anti-inductive  load  of  80  per  cent,  power-factor  ii  = 
-  0.67,  i  =  0.87: 


Cl  =  V4  X  106  -  4  72  -  0.5  7. 

As  seen,  due  to  the  internal  impedance,  and  especially  the 
resistance  of  this  machine,  the  regulation  is  very  poor,  and  even 
at  the  chosen  anti-inductive  load  no  rise  of  voltage  occurs. 

122.  Of  more  theoretical  interest  is  the  case  (b),  where  the 


206  ELECTRICAL  APPARATUS 

exciter  is  a  synchronous  motor,  and  the  synchronous-induction 
generator  produces  power  in  the  stator  and  in  the  rotor  circuit. 
In  this  case,  the  power  is  produced  by  the  generated  e.m.f.,  E 
(e.m.f.  of  mutual  induction,  or  of  the  rotating  magnetic  field), 
of  the  induction  machine,  and  energy  flows  outward  in  both 
circuits,  in  the  stator  into  the  receiving  circuit,  of  terminal 
voltage,  Ei,  in  the  rotor  against  the  impressed  e.m.f.  of  the 
synchronous  motor  exciter,  EQ.  The  voltage  of  one  receiving 
circuit,  the  stator,  therefore,  is  controlled  by  a  voltage  impressed 
upon  another  receiving  circuit,  the  rotor,  and  this  results  in  some 
interesting  effects  in  voltage  regulation. 

Assume  the  voltage,  E0,  impressed  upon  the  rotor  circuit  as 
the  nominal  generated  e.m.f.  of  the  synchronous-motor  exciter, 
that  is,  the  field  corresponding  to  the  exciter  field  -excitation, 
and  assume  the  field  excitation  of  the  exciter,  and  therewith 
the  voltage,  E0,  to  be  maintained  constant. 

Reducing  all  the  voltages  to  the  stator  circuit  by  the  ratio  of 
their  effective  turns  and  the  ratio  of  their  respective  frequencies, 
the  same  e.m.f.,  E,  is  generated  in  the  rotor  circuit  as  in  the 
stator  circuit  of  the  induction  machine. 

At  no-load,  neglecting  the  exciting  current  of  the  induction 
machine,  that  is,  with  no  current,  we  have  E0  =  E  =  EI. 

If  a  load  is  put  on  the  stator  circuit  by  taking  a  current,  7, 
from  the  same,  the  terminal  voltage,  Ei,  drops  below  the  gene- 
rated e.m.f.,  E,  by  the  drop  of  voltage  in  the  impedance,  Zi,  of 
the  stator  circuit.  Corresponding  to  the  stator  current,  /i,  a 
current,  72,  then  exists  in  the  rotor  circuit,  giving  the  same 
ampere-turns  as  Ii,  in  opposite  direction,  and  so  neutralizing  the 
m.m.f.  of  the  stator  (as  in  any  transformer).  This  current,  72, 
exists  in  the  synchronous  motor,  and  the  synchronous  motor 
e.m.f.,  #0,  accordingly  drops  below  the  generated  e.m.f.,  E,  of 
the  rotor,  or,  since  E0  is  maintained  constant,  E  rises  above  E0 
with  increasing  load,  by  the  drop  of  voltage  in  the  rotor  impedance, 
Zt,  and  the  synchronous  impedance,  Z0,  of  the  exciter. 

That  is,  the  stator  terminal  voltage,  Ei,  drops  with  increasing 
load,  by  the  stator  impedance  drop,  and  rises  with  increasing 
load  by  the  rotor  and  exciter  impedance  drop,  since  the  latter 
causes  the  generated  e.m.f.,  E,  to  rise. 

If  then  the  impedance  drop  in  the  rotor  circuit  is  greater  than 
that  in  the  stator,  with  increasing  load  the  terminal  voltage, 
EI,  of  the  machine  rises,  that  is,  the  machine  automatically 


SYNCHRONOUS  INDUCTION  GENERATOR       207 

over-compounds,  at  constant-exciter  field  excitation,  and  if  the 
stator  and  the  rotor  impedance  drops  are  equal,  the  machine 
compounds  for  constant  voltage. 

In  such  a  machine,  by  properly  choosing  the  stator  and  rotor 
impedances,  automatic  rise,  decrease  or  constancy  of  the  terminal 
voltage  with  the  load  can  be  produced. 

This,  however,  applies  only  to  non-inductive  load.  If  the 
current,  7,  differs  in  phase  from  the  generated  e.m.f.,  E,  the 
corresponding  current,  72,  also  differs;  but  a  lagging  component 
of  7i  corresponds  to  a  leading  component  in  72,  since  the  stator 
circuit  slips  behind,  the  rotor  circuit  is  driven  ahead  of  the 
rotating  magnetic  field,  and  inversely,  a  leading  component  of 
7i  gives  a  lagging  component  of  72.  The  reactance  voltage  of 
the  lagging  current  in  one  circuit  is  opposite  to  the  reactance 
voltage  of  the  leading  current  in  the  other  circuit,  therefore 
does  not  neutralize  it,  but  adds,  that  is,  instead  of  compounding, 
regulates  in  the  wrong  direction. 

123.  The  automatic  compounding  of  the  synchronous  induc- 
tion generator  with  low-frequency  synchronous-motor  excitation 
so  fails  if  the  load  is  not  non-inductive. 

Let: 

Zi  =  7*1  +  jxi  =  stator  self-inductive  impedance, 

Z%  =  r2  +  jx%  =  rotor  self-inductive  impedance,  reduced  to  the 

stator  circuit  by  the  ratio  of  the  effective  turns,  t  =  — ,  and  the 

ratio  of  frequencies,  a  =  •/ ', 

Ji 

ZQ  =  TQ  +  jx0  =  synchronous  impedance  of  the  synchronous- 
motor  exciter; 

EI  =  terminal  voltage  of  the  stator,  chosen  as  real  axis,  =  e\\ 

E0  =  nominal    generated    e.m.f.    of    the    synchronous-motor 
exciter,  reduced  to  the  stator  circuit; 

E  =  generated  e.m.f.  of  the  synchronous-induction  generator 
stator  circuit,  or  the  rotor  circuit  reduced  to  the  stator  circuit. 

The  actual  e.m.f.  generated  in  the  rotor  circuit  then  is  E'  = 
taE,  and  the  actual  nominal  generated  e.m.f.  of  the  synchronous 
exciter  is  E'0  =  taE0. 
Let: 

7i  =  i  —  jii  =  current  in  the   stator   circuit,  or  the  output 
current  of  the  machine. 


208  ELECTRICAL  APPARATUS 

The  current  in  the  rotor  circuit,  in  which  the  direction  of 
rotation  is  opposite,  or  ahead  of  the  revolving  field,  then  is, 
when  neglecting  the  exciter  current  : 


(If  F  =  exciting  admittance,  the  exciting  current  is  70  =  EY, 
and  the  total  rotor  current  then  70  +  1  2-) 
Then  in  the  rotor  circuit: 

E    =  E0  +  (Z0  +  Z2)  72,  (1) 

and  in  the  stator  circuit: 

E    =  Ei  +  Zi/L  (2) 

Hence: 

#1  =  #o  +  /2  (Z0  +  Z2)  -  ZiZi,  (3) 

or,  substituting  for  /i  and  72: 

tf  i  =  E0  +  *  (Z0  +  Z2  -  ZO  +  >'i  (Z0  +  Z2  +  Z,).       (4) 
Denoting  now: 

Z0  +  Zi  +  Z2  =  Z3  =  r3  4-  jz3,  /K\ 

Z0  +'Z2  *Zi-Z4-  r4+jx4, 

and  substituting: 

^   =  ^o  +  tZ4  +  jiiZa,  (6) 

or,  since  EI  =  e\\ 


(7) 
or  the  absolute  value: 

e02  =  Oi  -  r4t  +  X3ii)2  +  (^  +  r3ii)2.  (8) 

Hence: 


(9) 

That  is,  the  terminal  voltage,  ei,  decreases  due  to  the  decrease 
of  the  square  root,  but  may  increase  due  to  the  second  term. 
At  no-load: 

i    =  0,  i\  =  0  and  e\  =  e0. 


SYNCHRONOUS  INDUCTION  GENERATOR        209 

At  non-inductive  load: 

ii  =  0  and  ei  =  Ve02  -  xfi2  +  r&.  (10) 

ei  first  increases,  from  its  no-load  value,  e0,  reaches  a  maximum, 

and  then  decreases  again. 

Since: 

r4  =  r0  -f  r2  —  ri, 

#4  =  z0  +  #2  —  ri, 

at :  r4  =  0  and  #4  =  0, 

or, 

ri  =  r0  +  7*2, 
£1  =  £o  +  a;2, 
and: 

61  =60,  that  is,  in  this  case  the  terminal  vol- 
tage is  constant  at  all  non-inductive  loads,  at  constant  exciter 
excitation. 

In  general,  or  for  I\  =  i  —  jii, 

if  ii  is  positive  or  inductive  load,  from  equation  (9)  follows 
that  the  terminal  voltage,  d,  drops  with  increasing  load;  while 

if  ii  is  negative  or  anti-inductive  load,  the  terminal  voltage, 
ei,  rises  with  increasing  load,  ultimately  reaches  a  maximum 
and  then  decreases  again. 

From  equation  (9)  follows,  that  by  changing  the  impedances, 
the  amount  of  compounding  can  be  varied.  For  instance,  at 
non-inductive  load,  or  in  equation  (10)  by  increasing  the  re- 
sistance, r4,  the  voltage,  ci,  increases  faster  with  the  load. 

That  is,  the  overcompounding  of  the  machine  can  be  increased 
by  inserting  resistance  in  the  rotor  circuit. 

124.  As  an  example  is  shown,  in  Fig.  65,  in  full  line,  with  the 
total  current,  /  =  \/i2  -f-  i^,  as  abscissae,  the  voltage  regulation 
of  such  a  machine,  or  the  terminal  voltage,  ei,  with  a  four- 
cycle synchronous  motor  as  exciter  of  a  60-cycle  synchronous- 
induction  generator  driven  at  64-cycles  speed. 

1.  For  non-inductive  load,  or  /i  =  i.  (Curve  I.) 

2.  For    inductive    load    of    80    per    cent,    power-factor;    or 
1 1  =  I  (0.8  -  0.6  j).  (Curve  II.) 

3.  For  anti-inductive  load  of  80  per  cent,  power-factor;  or 
7i  =  /  (0.8+  0.6  j).  (Curve  III.) 

14 


210  -ELECTRICAL  APPARATUS 

For  the  constants: 


e0   =  2000  volts. 
Zl  =  0.1  +  0.3J. 
Z2=  1  +  0.5J. 


Z0  =  0.5  +  0.5  j. 
a  =  0.067. 

t  =  1,    that     is,     the 
same  number  of 
turns    in    stator 
.and  rotor. 
Then: 

Z3  =  1.6  +  1.3  j  and  Z4  =  1.4  +  0.7  j. 
Hence,  substituting  in  equation  (9) : 

e^  =  \/4  X  106  -  (0.7  i  +  1.6  z'i)2  +  1.4  i  -  1.3  n; 


2000 
2400 
2200 
2000 
1800 
1000 
1400 
1200 

I 

-91 

, 



—  -_ 

-^ 

s/ 

-^ 

X 

N 

y 

^ 

s 

V 

X 

x 

X 

X 

X 

N 

x 

x^ 

^••^ 

X 

^  —  . 

^, 

,—  - 

.-^ 

^ 

-*- 

LEAD 

I 

M- 

^- 

3-80-70-60-50-40-30-20-10  0   10  20  30  40  50  60  70  80  9 

FIG.  66. — Synchronous  induction  generator,  voltage  regulation  with  power- 
factor  of  load. 

thus,  for  non-inductive  load,  i\  =  0: 

d  =  A/4  X  106  -  0.49  i2  +  1.4  i; 
for  inductive  load  of  80  per  cent,  power-factor  ii  =  0.6  7;  i  =  0.8  7: 


X  106  -  2.31 12  +  0.34  /; 

and  for  anti-inductive  load  of  80  per  cent,  power-factor  ii  = 
-0.6  /;  i  =  0.87: 


6l  =  V±X  106  -  0.16  72  +  1.9  7. 

Comparing  the  curves  of  this  example  with  those  of  the  same 
machine  driven  as  frequency  converter  with  exciter  generator, 


SYNCHRONOUS  INDUCTION  GENERATOR   211 

and  shown  in.  dotted  lines  in  the  same  chart  (Fig.  65),  it  is 
seen  that  the  voltage  is  maintained  at  load  far  better,  and 
especially  at  inductive  load  the  machine  gives  almost  perfect 
regulation  of  voltage,  with  the  constants  assumed  here. 

To  show  the  variation  of  voltage  with  a  change  of  power- 
factor,  at  the  same  output  in  current,  in  Fig.  66,  the  terminal 
voltage,  ei,  is  plotted  with  the  phase  angle  as  abscissae,  from 
wattless  anti-inductive  load,  or  90°  lead,  to  wattless  inductive 
load,  or  90°  lag,  for  constant  current  output  of  400  amp.  As 
seen,  at  wattless  load  both  machines  give  the  same  voltage  but 
for  energy  load  the  type  (6)  gives  with  the  same  excitation  a 
higher  voltage,  or  inversely,  for  the  same  voltage  the  type  (a) 
requires  a  higher  excitation.  It  is,  however,  seen  that  with  the 
same  current  output,  but  a  change  of  power-factor,  the  voltage 
of  type  (a)  is  far  more  constant  in  the  range  of  inductive  load, 
while  that  of  type  (b)  is  more  constant  on  anti-inductive  load, 
and  on  inductive  load  very  greatly  varies  with  a  change-  of 
power-factor. 


CHAPTER  XIV 
PHASE  CONVERSION  AND  SINGLE-PHASE  GENERATION 

125.  Any  polyphase  system  can,  by  means  of  two  stationary 
transformers,  be  converted  into  any  other  polyphase  system, 
and  in  such  conversion,  a  balanced  polyphase  system  remains 
balanced,  while  an  unbalanced  system  converts  into  a  polyphase 
system  of  the  same  balance  factor.1 

In  the  conversion  between  single-phase  system  and  polyphase 
system,  a  storage  of  energy  thus  must  take  place,  as  the  balance 
factor  of  the  single-phase  system  is  zero  or  negative,  while  that 
of  the  balanced  polyphase  system  is  unity.  For  such  energy 
storage  may  be  used  capacity,  or  inductance,  or  momentum  or  a 
combination  thereof: 

Energy  storage  by  capacity,  that  is,  in  the  dielectric  field, 
required  per  kilo  volt-ampere  at  60  cycles  about  2000  c.c.  of 
space,  at  a  cost  of  about  $10.  Inductance,  that  is,  energy 
storage  by  the  magnetic  field,  requires  about  1000  c.c.  per  kilo- 
volt-ampere  at  60  cycles,  at  a  cost  of  $1,  while  energy  storage  by 
momentum,  as  kinetic  mechanical  energy,  assuming  iron  moving 
at  30  meter-seconds,  stores  1  kva.  at  60  cycles  by  about  3  c.c., 
at  a  cost  of  0.2c.,  thus  is  by  far  the  cheapest  and  least  bulky 
method  of  energy  storage.  Where  large  amounts  of  energy  have 
to  be  stored,  for  a  very  short  time,  mechanical  momentum  thus 
is  usually  the  most  efficient  and  cheapest  method. 

However,  size  and  cost  of  condensers  is  practically  the  same 
for  large  as  for  small  capacities,  while  the  size  and  cost  of  induc- 
tance decreases  with  increasing,  and  increases  with  decreasing 
kilovolt-ampere  capacity.  Furthermore,  the  use  of  mechanical 
momentum  means  moving  machinery,  requiring  more  or  less 
attention,  thus  becomes  less  suitable,  for  smaller  values  of  power. 
Hence,  for  smaller  amounts  of  stored  energy,  inductance  and 
capacity  may  become  more  economical  than  momentum,  and 
for  very  small  amounts  of  energy,  the  condenser  may  be  the 
cheapest  device.  The  above  figures  thus  give  only  the  approxi- 

l"  Theory  and  Calculation  of  Alternating-current  Phenomena,"  5th 
edition,  Chapter  XXXII. 

212 


PHASE  CONVERSION  213 

mate  magnitude  for  medium  values  of  energy,  and  then  apply 
only  to  the  active  energy-storing  structure,  under  the  assumption, 
that  during  every  energy  cycle  (or  half  cycle  of  alternating  cur- 
rent and  voltage),  the  entire  energy  is  returned  and  stored  again. 
While  this  is  the  case  with  capacity  and  inductance,  when  using 
momentum  for  energy  storage,  as  flywheel  capacity,  the  energy 
storage  and  return  is  accomplished  by  a  periodic  speed  variation, 
thus  only  a  part  of  the  energy  restored,  and  furthermore,  only 
a  part  of  the  structural  material  (the  flywheel,  or  the  rotor  of 
the  machine)  is  moving.  Thus  assuming  that  only  a  quarter 
of  the  mass  of  the  mechanical  structure  (motor,  etc.)  is  revolving, 
and  that  the  energy  storage  takes  place  by  a  pulsation  of  speed  of 
1  per  cent.,  then  1  kva.  at  60  cycles  would  require  600  c.c.  of 
material,  at  40c. 

Obviously,  at  the  limits  of  dielectric  or  magnetic  field  strength, 
or  at  the  limits  of  mechanical  speeds,  very  much  larger  amounts 
of  energy  per  bulk  could  be  stored.  Thus  for  instance,  at  the 
limits  of  steam-turbine  rotor  speeds,  about  400  meter-seconds, 
in  a  very  heavy  material  as  tungsten,  1  c.c.  of  material  would 
store  about  200  kva.  of  60-cycle  energy,  and  the  above  figures 
thus  represent  only  average  values  under  average  conditions. 

126.  Phase  conversion  is  of  industrial  importance  in  changing 
from  single-phase  to  polyphase,  and  in  changing  from  polyphase 
to  single-phase. 

Conversion  from  single-phase  to  polyphase  has  been  of  con- 
siderable importance  in  former  times,  when  alternating-current 
generating  systems  were  single-phase,  and  alternating-current 
motors  required  polyphase  for  their  operation.  With  the  prac- 
tically universal  introduction  of  three-phase  electric  power 
generation,  polyphase  supply  is  practically  always  available  for 
stationary  electric  motors,  at  least  motors  of  larger  size,  and 
conversion  from  single-phase  to  polyphase  thus  is  of  importance 
mainly : 

(a)  To  supply  small  amounts  of  polyphase  current,  for  the 
starting  of  smaller  induction  motors  operated  on  single-phase 
distribution    circuits,    2300    volts    primary,    or    110/220    volts 
secondary,  that  is,  in  those  cases,  in  which  the  required  amount 
of  power  is  not  sufficient  to  justify  bringing  the  third  phase  to 
the  motor:  with  larger  motors,  all  the  three  phases  are  brought 
to  the  motor  installation,  thus  polyphase  supply  used. 

(b)  For  induction-motor  railway  installations,  to  avoid  the 


214  ELECTRICAL  APPARATUS 

complication  and  inconvenience  incident  to  the  use  of  two  trolley 
wires.  In  this  case,  as  large  amounts  of  polyphase  power  are 
required,  and  economy  in  weight  is  important,  momentum  is 
generally  used  for  energy  storage,  that  is  an  induction  machine 
is  employed  as  phase  converter,  and  then  is  used  either  in  series 
or  in  shunt  to  the  motor. 

For  the  small  amounts  of  power  required  by  use  (a),  generally 
inductance  or  capacity  are  employed,  and  even  then  usually  the 
conversion  is  made  not  to  polyphase,  but  to  monocyclic,  as  the 
latter  is  far  more  economical  in  apparatus. 

Conversion  from  polyphase  to  single-phase  obviously  means 
the  problem  of  deriving  single-phase  power  from  a  balanced 
polyphase  system.  A  single-phase  load  can  be  taken  from  any 
phase  of  a  polyphase  system,  but  such  a  load,  when  consider- 
able, unbalances  the  polyphase  system,  that  is,  makes  the  vol- 
tages of  the  phases  unequal  and  lowers  the  generator  capacity. 
The  problem  thus  is,  to  balance  the  voltages  and  the  reaction  of 
the  load  on  the  generating  system. 

This  problem  has  become  of  considerable  importance  in  the 
last  years,  for  the  purpose  of  taking  large  single-phase  loads,  for 
electric  railway,  furnace  work,  etc.,  from  a  three-phase  supply 
system  as  a  central  station  or  transmission  line.  For  this  pur- 
pose, usually  synchronous  phase  converters  with  synchronous 
phase  balancers  are  used. 

As  illustration  may  thus  be  considered  in  the  following  the 
monocyclic  device,  the  induction  phase  converter,  and  the 
synchronous  phase  converter  and  balancer. 

Monocyclic  Devices 

127.  The  name  "monocyclic"  is  applied  to  a  polyphase  sys- 
tem of  voltages  (whether  symmetrical  or  unsymmetrical),  in 
which  the  flow  of  energy  is  essentially  single-phase. 

For  instance,  if,  as  shown  diagrammatically  in  Fig.  67,  we 
connect,  between  single-phase  mains,  AB,  two  pairs  of  non-in- 
ductive resistances,  r,  and  inductive  reactances,  x  (or  in  general, 
two  pairs  of  impedances  of  different  inductance  factors),  such 
that  r  =  x,  consuming  the  voltages  E\  and  Ez  respectively,  then 
the  voltage  eo  =  CD  is  in  quadrature  with,  and  equal  to,  the 
voltage  e  =  AB,  and  the  two  voltages,  e  and  eo,  constitute  a 
monocyclic  system  of  quarter-phase  voltages :  e  gives  the  energy 


,     PHASE  CONVERSION  215 

axis  of  the  monocyclic  system,  and  e0  the  quadrature  or  wattless 
axis.  That  is,  from  the  axis,  e,  power  can  be  drawn,  within 
the  limits  of  the  power-generating  system  back  of  the  supply 
voltage.  If,  however,  an  attempt  is  made  to  draw  power  from 
the  monocyclic  quadrature  voltage,  e0,  this  voltage  collapses. 

If  then  the  two  voltages,  e  and  e0,  are  impressed  upon  a  quarter- 
phase  induction  motor,  this  motor  will  not  take  power  equally 
from  both  phases,  e  and  eo,  but  takes  power  essentially  only  from 
phase,  e.  In  starting,  and  at  heavy  load,  a  small  amount  of 
power  is  taken  also  from  the  quadrature  voltage,  eo,  but  at  light- 
load,  power  may  be  returned  into  this  voltage,  so  that  in  general 
the  average  power  of  e0  approximates  zero,  that  is,  the  voltage, 
eo,  is  wattless. 

A  monocyclic  system  thus  may  be  defined  as  a  system  of  poly- 
phase voltages,  in  which  one  of  the  power  axis,  the  main  axis 
or  energy  axis,  is  constant  potential,  and  the  other  power  axis, 
the  auxiliary  or  quadrature  axis,  is  of  dropping  characteristic 
and  therefore  of  limited  power.  Or  it  may  be  defined  as  a  poly- 
phase system  of  voltage,  in  which  the  power  available  in  the  one 
power  axis  of  the  system  is  practically  unlimited  compared  with 
that  of  the  other  power  axis. 

A  monocyclic  system  thus  is  a  system  of  polyphase  voltage, 
which  at  balanced  polyphase  load  becomes  unbalanced,  that  is, 
in  which  an  unbalancing  of  voltage  or  phase  relation  occurs 
when  all  phases  are  loaded  with  equal  loads  of  equal  inductance 
factors. 

In  some  respect,  all  methods  of  conversion  from  single-phase 
to  polyphase  might  be  considered  as  monocyclic,  in  so  far  as  the 
quadrature  phase  produced  by  the  transforming  device  is  limited 
by  the  capacity  of  the  transforming  device,  while  the  main 
phase  is  limited  only  by  the  available  power  of  the  generating 
system.  However,  where  the  power  available  in  the  quadrature 
phase  produced  by  the  phase  converter  is  sufficiently  large  not 
to  constitute  a  limitation  of  power  in  the  polyphase  device  sup- 
plied by  it,  or  in  other  words,  where  the  quadrature  phase  pro- 
duced by  the  phase  converter  gives  essentially  a  constant-poten- 
t:al  voltage  under  the  condition  of  the  use  of  the  device,  then  the 
system  is  not  considered  as  monocyclic,  but  is  essentially 
polyphase. 

In  the  days  before  the  general  introduction  of  three-phase 
power  generation,  about  20  years  ago,  monocyclic  systems  were 


216  ELECTRICAL  APPARATUS 

extensively  used,  and  monocyclic  generators  built.  These  were 
single-phase  alternating-current  generators,  having  a  small 
quadrature  phase  of  high  inductance,  which  combined  with  the 
main  phase  gives  three-phase  or  quarter-phase  voltages.  The 
auxiliary  phase  was  of  such  high  reactance  as  to  limit  the  quadra- 
ture power  and  thus  make  the  flow  of  energy  essentially  single- 
phase,  that  is,  monocyclic.  The  purpose  hereof  was  to  permit 
the  use  of  a  small  quadrature  coil  on  the  generator,  and  thereby 
to  preserve  the  whole  generator  capacity  for  the  single-phase 
main  voltage,  without  danger  of  overloading  the  quadrature 
phase  in  case  of  a  high  motor  load  on  the  system.  The  general 
introduction  of  the  three-phase  system  superseded  the  mono- 
cyclic  generator,  and  monocyclic  devices  are  today  used  only 
for  local  production  of  polyphase  voltages  from  single-phase 
supply,  for  the  starting  of  small  single-phase  induction  motors, 
etc.  The  advantage  of  the  monocyclic  feature  then  consists  in 
limiting  the  output  and  thereby  the  size  of  the  device,  and  making 
it  thereby  economically  feasible  with  the  use  of  the  rather  expen- 
sive energy-storing  devices  of  inductance  (and  capacity)  used  in 
this  case. 

The  simplest  and  most  generally  used  monocyclic  device  con- 
sists of  two  impedances,  Zi  and  Z2,  of  different  inductance  factors 
(resistance  and  inductance,  or  inductance  and  capacity),  con- 
nected across  the  single-phase  mains,  A  and  B.  The  common 
connection,  C,  between  the  two  impedances,  Z\  and  Z2,  then  is  dis- 
placed in  phase  from  the  single-phase  supply  voltage,  A  and  B, 
and  gives  with  the  same  a  system  of  out-of-phase  voltages,  AC, 
CB  and  A  B,  or  a — more  or  less  unsymmetrical — three-phase 
triangle.  Or,  between  this  common  connection,  C,  and  the 
middle,  D,  of  an  autotransformer  connected  between  the  single- 
phase  mains,  AB,  a  quadrature  voltage,  CD,  is  produced. 

This  "manocyclic  triangle"  ACB,  in  its  application  as  single- 
phase  induction  motor-starting  device,  is  discussed  in  Chapter  V. 
Two  such  monocyclic  triangles  combined  give  the  monocyclic 
square,  Fig.  67. 

128.  Let  then,  in  the  monocyclic  square  shown  diagrammatic- 
ally  in  Fig.  67: 

7]  =  gl  —  jbl  =  admittance  AC  and  DB; 

Y2  =  02  —  jbz  =  admittance  CB  and  AD; 
and  let: 

YQ  =  gQ  —  jbo  =  admittance  of  the  load   on 


PHASE  CONVERSION 


217 


the  monocyclic  quadrature  voltage,  EQ  =  CD,  and  current,  70. 
Denoting  then : 

E  —  e  =  supply  voltage,  ABy  and  I  =  supply  current,  and 
EI,  EI  =  voltages,  I],  1 2  =  currents  in  the  two  sides  of  the 
monocyclic  square. 

It  is  then,  counting  voltages  and  currents  in  the  direction 
indicated  by  the  arrows  in  Fig.  67 : 


hence: 


and: 


substituting : 


into  (3)  gives: 


E2  +  E!  -  e,                                           .. 

T?             7?             1?    •  \                                                         \    ' 

C/2    —    VI    —    VO,  J 

777       |         777 

£/    -f-   £/Q 

2 

e  -  EQ. 

(2) 

2 

/      =/l+/2, 

/o  =  /,  -  /.; 

(3) 

/TJT  rr 
0    —    -C/0-t  0> 

7                       T7T      V" 

/  1    —    &\1   1, 

(4) 

/2  =  E*Y2', 

(5) 


substituting  (2)  into  (5)  gives: 


»  e  l/i  --  12;     . 

'°  "  F!  +  F,  +  2  Fo' 


(6) 


substituting  (6)  into  (2)  gives: 


(F!  +  Fo) 
+  F2  +  2F0' 


(7) 


218 


ELECTRICAL  APPARATUS 


substituting  (7)  and  (6)  into  (4)  and  (5)  gives  the  currents: 

£>V    (V      -  V  \ 

r          ex  Q(I  i  —  1 2)   _ 


= 

•f 


F!  +  F2  +  2  Fo 
e  (FoFi  +  F0F2  - 


+  F2  +  2  F 


(F 


F!  +  F2  +  2  Fo 
'  eF2  (F!  +  Fo) 

"  F,  +  F2  +  2  Fo 
129.  For  a  combination  of  equal  resistance  and  reactance : 


(8) 


RESISTANCE-INDUCTANCE    // 

MONOCYCLIC  SQUARE 
T  =  X  =7.07     OHMS 

e  =  100  VOLTS 

E, 


FIG.  67.  —  Resistance-inductance  monocyclic  square,  topographical  regula- 
tion characteristic. 


=  -jo; 


and  a  load  : 


Fo  =  a(p  -  jq); 
equations  (6)  and  (8)  give: 


1  -  j  +  2  (p  -  jq) 


T      = 

" 


ea        ~ 


1  -  j  +  2  (p  -  j 


PHASE  CONVERSION 


219 


Fig.  67  shows  the  voltage  diagram,  and  Fig.  68  the  regulation, 
that  is,  the  values  of  eQ  and  i,  with  IQ  as  abscissae,  for: 

e  =  100  volts, 
a  =  0.1  \/2  mho. 


\ 

^ 

AMP. 

19 

\ 

\ 

R 

REGULATION  OF 
ESISTANCE-INDUCTANC 
MONOCYCLIC  SQUARE 
T-X  -7.07  OHMS 

e  =  100  VOLTS 

;E 

_18. 
_17_ 
16 

S 

V 

X 

\ 

\ 

\ 

15 

\ 

\ 

14 

\ 

\ 

13 

\ 

\ 

12 

\ 

11 

e 

\ 

VOLTS 

100 

_10. 

s 

x, 

90 

§ 

\ 

80 

\ 

X° 

70 

e\; 

\ 

60 

\ 

\ 

50 

\ 

\ 

40 

X 

\ 

30 

<*'* 

V 

\ 

20 

^x" 

^ 

•^J 

7 

^v^ 

^ 

\ 

10 

^ 

x^ 

! 

» 

\ 

AMP. 

1 

1 

—  >• 

1 

: 

^^^ 

| 

"S 

10 

FIG.  68. — Resistance-inductance  monocyclic  square,  regulation  curve. 

For:  q  =  0,  that  is,  non-inductive  load,  the  voltage  diagram 
is  a  curve  shown  by  circles  in  Fig.  67,  for  0,  2,  4,  6,  8  and  10  amp. 
load,  the  latter  being  the  maximum  or  short-circuit  value. 

For  q  =  p,  or  a  load  of  45°  load,  the  voltage  diagram  is  the 
straight  line  shown  by  crosses  in  Fig.  67.  That  is,  in  this  case, 
the  monocyclic  voltage,  e0,  is  in  quadrature  with  the  supply  voltage, 


220  ELECTRICAL  APPARATUS 

e,  at  all  loads,  while  for  non-inductive  load  the  monocyclic  voltage, 
Co,  not  only  shrinks  with  increasing  load,  but  also  shifts  in  phase, 
from  quadrature  position,  and  the  diagram  is  in  the  latter  case 
shown  for  4  amp.  load  by  the  dotted  lines  in  Fig.  67. 

In  Fig.  68  the  drawn  lines  correspond  to  non-inductive  load. 
The  regulation  for  45°  lagging  load  is  shown  by  dotted  lines  in 
Fig.  68. 

e'0  shows  the  quadrature  component  of  the  monocyclic  voltage, 
Co,  at  non-inductive  load.  That  is,  the  component  of  Co,  which  is 
in  phase  with  e,  and  therefore  could  be  neutralized  by  inserting 
into  e0  a  part  of  the  voltage,  e,  by  transformation. 

As  seen  in  Fig.  68,  the  supply  current  is  a  maximum  of  20  amp. 
at  no-load,  and  decreases  with  increasing  load,  to  10  amp.  at 
short-circuit  load. 

The  apparent  efficiency  of  the  device,  that  is,  the  ratio  of  the 
volt-ampere  output: 

QQ  =  ColQ 

to  the  volt-ampere  input: 

Q  =  ei 

is  given  by  the  curve,  7,  in  Fig.  68. 

As  seen,  the  apparent  efficiency  is  very  low,  reaching  a  maxi- 
mum of  14  per  cent.  only. 

If  the  monocyclic  square  is  produced  by  capacity  and  induc- 
tance, the  extreme  case  of  dropping  of  voltage,  Co,  with  increase  of 
current,  i0j  is  reached  in  that  the  circuit  of  the  voltage,  e0,  becomes 
a  constant-current  circuit,  and  this  case  is  more  fully  discussed 
in  Chapter  XIV  of  " Theory  and  Calculation  of  Electric  Circuits" 
as  a  constant-potential  constant-current  transforming  device. 

Induction  Phase  Converter 

130.  The  magnetic  field  of  a  single-phase  induction  motor  at 
or  near  synchronism  is  a  uniform  rotating  field,  or  nearly  so, 
deviating  from  uniform  intensity  and  uniform  rotation  only  by 
the  impedance  drop  of  the  primary  winding.  Thus,  in  any  coil 
displaced  in  position  from  the  single-phase  primary  coil  of  the 
induction  machine,  a  voltage  is  induced  which  is  displaced  in 
phase  from  the  supply  voltage  by  the  same  angle  as  the  coil  is 
displaced  in  position  from  the  coil  energized  by  the  supply  vol- 
tage. An  induction  machine  running  at  or  near  synchronism 
thus  can  be  used  as  phase  converter,  receiving  single-phase  sup- 


PHASE  CONVERSION 


221 


ply  voltage,  E0,  and  current,  70,  in  one  coil,  and  producing  a  voltage 
of  displaced  phase,  E2)  and  current  of  displaced  phase,  72,  in 
another  coil  displaced  in  position. 

Thus  if  a  quarter-phase  motor  shown  diagrammatically  in  Fig. 
69A  is  operated  by  a  single-phase  voltage,  E0j  supplied  to  the  one 


Co 
ID 


Io 


It 


Yo 


FIG.  69. — Induction  phase  converter  diagram. 

phase,  in  the  other  phase  a  quadrature  voltage,  E2,  is  produced 
and  quadrature  current  can  be  derived  from  this  phase. 

The  induction  machine,  Fig.  69 A,  is  essentially  a  transformer, 
giving  two  transformations  in  series:  from  the  primary  supply 
circuit,  Eoloj  to  the  secondary  circuit  or  rotor,  E\I\t  and  from  the 
rotor  circuit,  E\I\,  as  primary  circuit,  to  the  other  stator  circuit 


222  ELECTRICAL  APPARATUS 

or  second  phase,  E2I2,  as  secondary  circuit.  It  thus  can  be  repre- 
sented diagrammatically  by  the  double  transformer  Fig.  69B. 

The  only  difference  between  Fig.  69A  and  695  is,  that  in  Fig. 
69 A  the  synchronous  rotation  of  the  circuit,  EJi,  carries  the  cur- 
rent, /i,  90°  in  space  to  the  second  transformer,  and  thereby  pro- 
duces a  90°  time  displacement.  That  is,  primary  current  and 
voltage  of  the  second  transformer  of  Fig.  69J5  are  identical  in 
intensity  with  the  secondary  currents  and  voltage  of  the  first 
transformer,  but  lag  behind  them  by  a  quarter  period  in  space 
and  thus  also  in  time.  The  momentum  of  the  rotor  takes  care 
of  the  energy  storage  during  this  quarter  period. 

As  the  double  transformer,  Fig.  695,  can  be  represented  by 
the  double  divided  circuit,  Fig.  69C,1  Fig.  69C  thus  represents 
the  induction  phase  converter,  Fig.  69 A,  in  everything  except 
that  it  does  not  show  the  quarter-period  lag. 

As  the  equations  derived  from  Fig.  69C  are  rather  complicated, 
the  induction  converter  can,  with  sufficient  approximation  for 
most  purposes,  be  represented  either  by  the  diagram  Fig.  69D, 
or  by  the  diagram  Fig.  69E.  Fig.  69Z)  gives  the  exciting  current 
of  the  first  transformer  too  large,  but  that  of  the  second  trans- 
former too  small,  so  that  the  two  errors  largely  compensate. 
The  reverse  is  the  case  in  Fig.  69^,  and  the  correct  value,  cor- 
responding to  Fig.  69C,  thus  lies  between  the  limits  69D  and  69£J. 
The  error  made  by  either  assumption,  69D  or  69#,  thus  must  be 
smaller  than  the  difference  between  these  two  assumptions. 

131.  Let: 

Fo  =  0o  —  jbQ  =  primary  exciting  admittance  of  the  induc- 
tion machine, 

ZQ  =  r0  -{-  j%o  =  primary,  and  thus  also  tertiary  self-induc- 
tive impedance, 

Zi  =  7*1  +  j%i  =  secondary  self-inductive  impedance, 
all  at  full  frequency,  and  reduced  to  the  same  number  of  turns. 
Let: 

Yz  =  </2  —  jb%  =  admittance  of  the  load  on  the  second  phase ; 
denoting  further: 

Zi    —   ZQ  +  Zit 

1  lt  Theory  and  Calculation  of  Alternating-current  Phenomena,"  5th 
edition,  page  204. 


PHASE  CONVERSION  223 

it  is,  then,  choosing  the  diagrammatic  representation,  Fig.  69D: 

7o  -  E0Y0  =  72  +  #2F0  =  /!,  (9) 

#o  =  #2  +  2Z(72  +  #2F0),  (10) 

h  =  #2F2;  (11) 


substituting  (11)  into  (10)  and  transposing,  gives: 

^2  =  1  +  2Z(F0+F2); 
if  the  diagram,  Fig.  69E,  is  used,  it  is: 


°  ' 


l+2Z(F0+F2[l  +  FoZ]) 

which  differs  very  little  from  (12). 
And,  substituting  (11)  and  (12)  into  (9): 

/O   =    #2  (F0   +    F2)    +  #0Fo, 

F2  +  2F0+2ZFo(Fo+F2) 


1+2Z(F0+  F2) 


(14) 


Equations  (11),  (12)  and  (13)  give  for  any  value  of  load,  F2, 
on  the  quadrature  phase,  the  values  of  voltage,  E2,  and  current, 
72,  of  this  phase,  and  the  supply  current,  70,  at  supply  voltage,  EQ. 

It  must  be  understood,  however,  that  the  actual  quadrature 
voltage  is  not  E^,  but  is  jE2,  carried  a  quarter  phase  forward  by 
the  rotation,  as  discussed  before. 

132.  As  instance,  consider  a  phase  converter  operating  at  con- 
stant supply  voltage: 

E0  =  e0  =  100  volts; 
of  the  constants: 

Fo  =  0.01  -  O.lj, 

Z0  =  Z1  =  0.05  +  0.15;; 

thus: 

Z  =  0.1  +  0.3J; 

and  let: 

F2  =  a  (p  -  jq) 
==  a  (0.8-  0.6  j), 

that  is,  a  load  of  80  per  cent,  power-factor,  which  corresponds 
about  to  the  average  power-factor  of  an  induction  motor. 


224  ELECTRICAL  APPARATUS 

It  is,  then,  substituted  into  (11)  to  (13): 

100 


#2  = 


(1.062  +  0.52  a)  +  j  (0.36  a  -  0.028) 

(80  -  60  j)  a         

(1.062 +  0.52  a)  +j(0.36a  -  0.028) ; 
for: 

a  =  0,  or  no-load,  this  gives: 

e2  =  94.1, 

ii  =  0, 

10  =  19.5; 
for: 

a  =  oo  >  or  short-circuit,  this  gives : 

e2  =  0, 
it  =  159, 

11  =  169. 

The  voltage  diagram  is  shown  in  Fig.  70,  and  the  load  char- 
acteristics or  regulation  curves  in  Fig.  71. 

As  seen:  the  voltage,  e2,  is  already  at  no-load  lower  than  the 
supply  voltage,  e0,  due  to  the  drop  of  voltage  of  the  exciting  cur- 
rent in  the  self-inductive  impedance  of  the  phase  converter. 

In  Fig.  70  are  marked  by  circles  the  values  of  voltage,  e2,  for 
every  20  per  cent,  of  the  short-circuit  current. 

Fig.  71  gives  the  quadrature  component  of  the  voltage,  e2,  as 
e"2,  and  the  apparent  efficiency,  or  ratio  of  volt-ampere  output 
to  volt-ampere  input : 

and  the  primary  supply  current,  io. 

It  is  interesting  to  compare  the  voltage  diagram  and  especially 
the  load  and  regulation  curves  of  the  induction  phase  converter, 
Figs.  70  and  71,  with  those  of  the  monocyclic  square,  Figs.  67 
and  68. 

As  seen,  in  the  phase  converter,  the  supply  current  at  no-load 
is  small,  is  a  mere  induction-machine  exciting  current,  and  in- 
creases with  the  load  and  approximately  proportional  thereto. 

The  no-load  input  of  both  devices  is  practically  the  same,  but 
the  voltage  regulation  of  the  phase  converter  is  very  much  better : 
the  voltage  drops  to  zero  at  159  amp.  output,  while  that  of  the 


PHASE  CONVERSION 


225 


INDUCTION   PHASE  CONVERTER 
Y0=.01-.1j, 


FIG.  70. — Induction  phase  converter,  topographic  regulation  characteristic. 


VOLTS 

150 


INDUCTION   PHASE  CONVERTER 
Y0=.01-  .1  3i    Z0=Z,=.05  +.15  j 

Y2=a  (.8-.6J) 

e0=100  VOLTS 


.140. 


.120. 
J.1Q. 

100 
90 


FIG.  71. — Induction  phase  converter,  regulation  curve. 


226  ELECTRICAL  APPARATUS 

monocyclic  square  reaches  zero  already  at  10  amp.  output.  This 
illustrates  the  monocyclic  character  of  the  latter,  that  is,  the  limi- 
tation of  the  output  of  the  quadrature  voltage. 

As  the  result  hereof,  the  phase  converter  reaches  fairly  good 
apparent  efficiencies,  54  per  cent.,  and  reaches  these  already  at 
moderate  loads. 

The  quadrature  component,  e"2,  of  the  voltage,  e0,  is  much 
smaller  with  the  phase  converter,  and,  being  in  phase  with  the 
supply  voltage,  e0,  can  be  eliminated,  and  rigid  quadrature  relation 
of  ez  with  e0  maintained,  by  transformation  of  a  voltage  —  e"2 
from  the  single-phase  supply  into  the  secondary.  Furthermore,  as 
e"2  is  approximately  proportional  to  iQ — except  at  very  low  loads 
— it  could  be  supplied  without  regulation,  by  a  series  transformer, 
that  is,  by  connecting  the  primary  of  a  transformer  in  series  with 
the  supply  circuit,  i0,  the  secondary  in  series  with  e2.  Thereby 
e2  would  be  maintained  in  almost  perfect  quadrature  relation 
to  eQ  at  all  important  loads. 

Thus  the  phase  converter  is  an  energy-transforming  device, 
while  the  monocyclic  square,  as  the  name  implies,  is  a  device  for 
producing  an  essentially  wattless  quadrature  voltage. 

133.  A  very  important  use  of  the  induction  phase  converter 
is  in  series  with  the  polyphase  induction  motor  for  which  it  sup- 
plies the  quadrature  phase. 

In  this  case,  the  phase,  e0,  io  of  the  phase  converter  is  connected 
in  series  to  one  phase,  eVo,  of  the  induction  motor  driving  the 
electric  car  or  polyphase  locomotive,  into  the  circuit  of  the  single- 
phase  supply  voltage,  e  =  e0  +  e'0,  and  the  second  phase  of  the 
phase  converter,  e2,  i2,  is  connected  to  the  second  phase  of  the 
induction  motor. 

This  arrangement  still  materially  improves  the  polyphase  regu- 
lation: the  induction  motor  receives  the  voltages: 

e'o  =  e  —  eo, 
and: 

e'2  =  e2. 

At  no-load,  e2  is  a  maximum.  With  increasing  load,  e2  =  e'2 
drops,  and  hereby  also  drops  the  other  phase  voltage  of  the  in- 
duction motor,  e'0.  This,  however,  raises  the  voltage,  eQ  =  e  — 
e'o,  on  the  primary  phase  of  the  phase  converter,  and  hereby 
raises  the  secondary  phase  voltage,  e2  =  e'2,  thus  maintains  the 


PHASE  CONVERSION  227 

two  voltages  e'0  and  e'2  impressed  upon  the  induction  motor  much 
more  nearly  equal,  than  would  be  the  case  with  the  use  of  the 
phase  converter  in  shunt  to  the  induction  motor. 

Series  connection  of  the  induction  phase  converter,  to  the  in- 
duction motor  supplied  by  it,  thus  automatically  tends  to  regu- 
late for  equality  of  the  two-phase  voltages,  e'o  and  e'2,  of  the  induc- 
tion motor.  Quadrature  position  of  these  two-phase  voltages 
can  be  closely  maintained  by  a  series  transformer  between  iQ  and 
i2,  as  stated  above. 

It  is  thereby  possible  to  secure  practically  full  polyphase  motor 
output  from  an  induction  motor  operated  from  single-phase  sup- 
ply through  a  series-phase  converter,  while  with  parallel  connec- 
tion of  the  phase  converter,  the  dropping  quadrature  voltage 
more  or  less  decreases  the  induction  motor  output.  For  this 
reason,  for  uses  where  maximum  output,  and  especially  maximum 
torque  at  low  speed  and  in  acceleration  is  required,  as  in  rail- 
roading, the  use  of  the  phase  converter  in  series  connections  to 
the  motor  is  indicated. 

Synchronous  Phase  Converter  and  Single -phase  Generation 

134.  While  a  small  amount  of  single-phase  power  can  be  taken 
from  a  three-phase  or  in  general  a  polyphase  system  without  dis- 
turbing the  system,  a  large  amount  of  single-phase  power  results 
in  unbalancing  of  the  three-phase  voltages  and  impairment  of 
the  generator  output. 

With  balanced  load,  the  impedance  voltages,  e'  =  iz,  of  a  three- 
phase  system  are  balanced  three-phase  voltages,  and  their  effect 
can  be  eliminated  by  inserting  a  three-phase  voltage  into  the 
system  by  three-phase  potential  regulator  or  by  increasing  the 
generator  field  excitation.  The  impedance  voltages  of  a  single- 
phase  load,  however,  are  single-phase  voltages,  and  thus,  com- 
bined with  the  three-phase  system  voltage,  give  an  unbalanced 
three-phase  system.  That  is,  in  general,  the  loaded  phase  drops 
in  voltage,  and  one  of  the  unloaded  phases  rises,  the  other  also 
drops,  and  this  the  more,  the  greater  the  impedance  in  the  circuit 
between  the  generated  three-phase  voltage  and  the  single-phase 
load.  Large  single-phase  load  taken  from  a  three-phase  trans- 
mission line — as  for  instance  by  a  supply  station  of  a  single-phase 
electric  railway — thus  may  cause  an  unbalancing  of  the  trans- 
mission-line voltage  sufficient  to  make  it  useless. 

A  single-phase  system  of  voltage,  e,  may  be  considered  as  com- 
bination of  two  balanced  three-phase  systems  of  opposite  phase 


228  ELECTRICAL  APPARATUS 

e    ee  ee2  e    ee2  ee  —  1  -f  j  A/3 

rotation:  ^  y  y  and  -^  -^  -^  where  e  =  i/1  =  -      — --- 

The  unbalancing  of  voltage  caused  by  a  single-phase  load  of 
impedance  voltage,  e  =  iz,  thus  is  the  same  as  that  caused  by 
two  three-phase  impedance  voltages,  e/2,  of  which  the  one  has 
the  same,  the  other  the  opposite  phase  rotation  as  the  three-phase 
supply  system.  The  former  can  be  neutralized  by  raising  the 
supply  voltage  by  e/2,  by  potential  regulator  or  generator  excita- 
tion. This  means,  regulating  the  voltage  for  the  average  drop. 
It  leaves,  however,  the  system  unbalanced  by  the  impedance 
voltage,  e/2,  of  reverse-phase  rotation.  The  latter  thus  can  be 
compensated,  and  the  unbalancing  eliminated,  by  inserting  into 
the  three-phase  system  a  set  of  three-phase  voltages,  e/2,  of  re- 
verse-phase rotation.  Such  a  system  can  be  produced  by  a  three- 
phase  potential  regulator  by  interchanging  two  of  the  phases. 
Thus,  if  A,  B,  C  are  the  three  three-phase  supply  voltages,  im- 
pressed upon  the  primary  or  shunt  coils  a,  b,  c  of  a  three-phase 
potential  regulator,  and  1,  2,  3  are  the  three  secondary  or  series 
coils  of  the  regulator,  then  the  voltages  induced  in  1,  3,  2  are 
three-phase  of  reverse-phase  rotation  to  A,  B,  C,  and  can  be  in- 
serted into  the  system  for  balancing  the  unbalancing  due  to 
single-phase  load,  in  the  resultant  voltage:  A  +  1,5  +  3,  C  +  2. 
It  is  obviously  necessary  to  have  the  potential  regulator  turned 
into  such  position,  that  the  secondary  voltages  1,  3,  2  have  the 
proper  phase  relation.  This  may  require  a  wider  range  of  turn- 
ing than  is  provided  in  the  potential  regulator  for  controlling 
balanced  voltage  drop. 

It  thus  is  possible  to  restore  the  voltage  balance  of  a  three- 
phase  system,  which  is  unbalanced  by  a  single-phase  load  of  im- 
pedance voltage,  e',  by  means  of  two  balanced  three-phase  poten- 
tial regulators  of  voltage  range,  e'/2,  connected  so  that  the  one 
gives  the  same,  the  other  the  reverse  phase  rotation  of  the  main 
three-phase  system. 

Such  an  apparatus  producing  a  balanced  polyphase  system  of 
reversed  phase  rotation,  for  inserting  in  series  into  a  polyphase 
system  to  restore  the  balance  on  single-phase  load,  is  called  a 
phase  balancer,  and  in  the  present  case,  a  stationary  induction 
phase  balancer. 

A  synchronous  machine  of  opposite  phase  rotation  to  the  main 
system  voltages,  and  connected  in  series  thereto,  would  then  be 
a  synchronous  phase  balancer, 


PHASE  CONVERSION  229 

The  purpose  of  the  phase  balancer,  thus,  is  the  elimination  of 
the  voltage  unbalancing  due  to  single-phase  load,  and  its  capacity 
must  be  that  of  the  single-phase  impedance  volt-amperes.  It 
obviously  can  not  equalize  the  load  on  the  phases,  but  the  flow 
of  power  of  the  system  remains  unbalanced  by  the  single-phase 
load. 

135.  The  capacity  of  large  synchronous  generators  is  essentially 
determined  by  the  heating  of  the  armature  coils.  Increased  load 
on  one  phase,  therefore,  is  not  neutralized  by  lesser  load  on  the 
other  phases,  in  its  limitation  of  output  by  heating  of  the  arma- 
ture coils  of  the  generators. 

The  most  serious  effect  of  unbalanced  load  on  the  generator  is 
that  due  to  the  pulsating  armature  reaction.  With  balanced 
polyphase  load,  the  armature  reaction  is  constant  in  intensity 
and  in  direction,  with  regards  to  the  field.  With  single-phase 
load,  however,  the  armature  reaction  is  pulsating  between  zero 
and  twice  its  average  value,  thus  may  cause  a  double-frequency 
pulsation  of  magnetic  flux,  which,  extending  through  the  field 
circuit,  may  give  rise  to  losses  and  heating  by  eddy  currents  in 
the  iron,  etc.  With  the  slow-speed  multipolar  engine-driven 
alternators  of  old,  due  to  the  large  number  of  poles  and  low  per- 
ipheral speed,  the  ampere-turns  armature  reaction  per  pole 
amounted  to  a  few  thousand  only,  thus  were  not  sufficient  to 
cause  serious  pulsation  in  the  magnetic-field  circuit.  With  the 
large  high-speed  turbo-alternators  of  today,  of  very  few  poles, 
and  to  a  somewhat  lesser  extent  also  with  the  larger  high-speed 
machines  driven  by  high-head  water  wheels,  the  armature  reac- 
tion per  pole  amounts  to  very  many  thousands  of  ampere-turns. 
Section  and  length  of  the  field  magnetic  circuit  are  very  large. 
Even  a  moderate  pulsation  of  armature  reaction,  due  to  the  un- 
balancing of  the  flow  of  power  by  single-phase  load,  then,  may 
cause  very  large  losses  in  the  field  structure,  and  by  the  resultant 
heating  seriously  reduce  the  output  of  the  machine. 

It  then  becomes  necessary  either  to  balance  the  load  between 
the  phases,  and  so  produce  the  constant  armature  reaction  of 
balanced  polyphase  load,  or  to  eliminate  the  fluctuation  of  the 
armature  reaction.  The  latter  is  done  by  the  use  of  an  effective 
squirrel-cage  short-circuit  winding  in  the  pole  faces.  The  double- 
frequency  pulsation  of  armature  reaction  induces  double-fre- 
quency currents  in  the  squirrel  cage — just  as  in  the  single-phase 
induction  motor — and  these  induced  currents  demagnetize,  when 


230  ELECTRICAL  APPARATUS 

the  armature  reaction  is  above,  and  magnetize  when  it  is  below 
the  average  value,  and  thereby  reduce  the  fluctuation,  that  is, 
approximate  a  constant  armature  reaction  of  constant  direction 
with  regards  to  the  field — that  is,  a  uniformly  rotating  magnetic 
field  with  regards  to  the  armature. 

However,  for  this  purpose,  the  m.m.f.  of  the  currents  induced 
in  the  squirrel-cage  winding  must  equal  that  of  the  armature 
winding,  that  is,  the  total  copper  cross-section  of  the  squirrel  cage 
must  be  of  the  same  magnitude  as  the  total  copper  cross-section 
of  the  armature  winding.  A  small  squirrel  cage,  such  as  is  suffi- 
cient for  starting  of  synchronous  motors  and  for  anti-hunting 
purposes,  thus  is  not  sufficient  in  high  armature-reaction  machines 
to  take  care  of  unbalanced  single-phase  load. 

A  disadvantage  of  the  squirrel-cage  field  winding,  however, 
is,  that  it  increases  the  momentary  short-circuit  current  of 
the  generator,  and  retards  its  dying  out,  therefore  increases  the 
danger  of  self-destruction  of  the  machine  at  short-circuit.  In 
the  first  moment  after  short-circuit,  the  field  poles  still  carry  full 
magnetic  flux — as  the  field  can  not  die  out  instantly.  No  flux 
passes  through  the  armature — except  the  small  flux  required  to 
produce  the  resistance  drop,  ir.  Thus  practically  the  total  field 
flux  must  be  shunted  along  the  air  gap,  through  the  narrow  sec- 
tion between  field  coils  and  armature  coils.  As  the  squirrel-cage 
winding  practically  bars  the  flux  to  cross  it,  it  thereby  further 
reduces  the  available  flux  section  and  so  increases  the  flux  density 
and  with  it  the  momentary  short-circuit  current,  which  gives 
the  m.m.f.  of  this  flux. 

It  must  also  be  considered  that  the  reduction  of  generator  out- 
put resulting  from  unequal  heating  of  the  armature  coils  due  to 
unequal  load  on  the  phases  is  not  eliminated  by  a  squirrel-cage 
winding,  but  rather  additional  heat  produced  by  the  currents 
in  the  squirrel-cage  conductors. 

136.  A  synchronous  machine,  just  as  an  induction  machine, 
may  be  generator,  producing  electric  power,  or  motor,  receiving 
electric  power,  or  phase  converter,  receiving  electric  power  in 
some  phase,  the  motor  phase,  and  generating  electric  power  in 
some  other  phase,  the  generator  phase.  In  the  phase  converter, 
the  total  resultant  armature  reaction  is  zero,  and  the  armature 
reaction  pulsates  with  double  frequency  between  equal  positive 
and  negative  values.  Such  phase  converter  thus  can  be  used  to 
produce  polyphase  power  from  a  single-phase  supply.  The  in- 


PHASE  CONVERSION  231 

duction  phase  converter  has  been  discussed  in  the  preceding,  and 
the  synchronous  phase  converter  has  similar  characteristics,  but 
as  a  rule  a  better  regulation,  that  is,  gives  a  better  constancy  of 
voltage,  and  can  be  made  to  operate  without  producing  lagging 
currents,  by  exciting  the  fields  sufficiently  high. 

However,  a  phase  converter  alone  can  not  distribute  single- 
phase  load  so  as  to  give  a  balanced  polyphase  system.  When 
transferring  power  from  the  motor  phase  to  the  generator  phase, 
the  terminal  voltage  of  the  motor  phase  equals  the  induced  vol- 
tage plus  the  impedance  drop  in  the  machine,  that  of  the  gen- 
erator phase  equals  induced  voltage  minus  the  impedance  drop, 
and  the  voltage  of  the  motor  phase  thus  must  be  higher  than  that 
of  the  generator  phase  by  twice  the  impedance  voltage  of  the 
phase  converter  (vectorially  combined). 

Therefore,  in  converting  single-phase  to  polyphase  by  phase 
converter,  the  polyphase  system  produced  can  not  be  balanced 
in  voltage,  but  the  quadrature  phase  produced  by  the  converter 
is  less  than  the  main  phase  supplied  to  it,  and  drops  off  the  more, 
the  greater  the  load. 

In  the  reverse  conversion,  however,  distributing  a  single-phase 
load  between  phases  of  a  polyphase  system,  the  voltage  of  the 
generator  phase  of  the  converter  must  be  higher,  that  of  the  motor 
phase  lower  than  that  of  the  polyphase  system,  and  as  the  gen- 
erator phase  is  lower  in  voltage  than  the  motor  phase,  it  follows, 
that  the  phase  converter  transfers  energy  only  when  the  poly- 
phase system  has  become  unbalanced  by  more  than  the  voltage 
drop  in  the  converter.  That  is,  while  a  phase  converter  may 
reduce  the  unbalancing  due  to  single-phase  load,  it  can  never 
restore  complete  balance  of  the  polyphase  system,  in  voltage  and 
in  the  flow  of  power.  Even  to  materially  reduce  the  unbalancing, 
requires  large  converter  capacity  and  very  close  voltage  regula- 
tion of  the  converter,  and  thus  makes  it  an  uneconomical  machine. 

To  balance  a  polyphase  system  under  single-phase  load,  there- 
fore, requires  the  addition  of  a  phase  balancer  to  the  phase 
converter.  Usually  a  synchronous  phase  balancer,  would  be 
employed  in  this  case,  that  is,  a  small  synchronous  machine  of 
opposite  phase  rotation,  on  the  shaft  of  the  phase  converter, 
and  connected  in  series  thereto.  Usually  it  is  connected  into 
the  neutral  of  the  phase  converter.  By  the  phase  balancer,  the 
voltage  of  the  motor  phase  of  the  phase  converter  is  raised 
above  the  generator  phase  so  as  to  give  a  power  transfer  sufficient 


232  ELECTRICAL  APPARATUS 

to  balance  the  polyphase  system,  that  is,  to  shift  half  of  the  single 
phase  power  by  a  quarter  period,  and  thus  produce  a  uniform 
flow  of  power. 

Such  synchronous  phase  balancer  constructively  is  a  synchro- 
nous machine,  having  two  sets  of  field  poles,  A  and  B,  in  quad- 
rature with  each  other.  Then  by  varying  or  reversing  the 
excitation  of  the  two  sets  of  field  poles,  any  phase  relation  of  the 
reversely  rotating  polyphase  system  of  the  balancer  to  that  of  the 
converter  can  be  produced,  from  zero  to  360°. 

137.  Large  single-phase  powers,  such  as  are  required  for  single- 
phase  railroading,  thus  can  be  produced. 

(a)  By  using  single-phase  generators  and  separate  single-phase 
supply  circuits. 

(b)  By  using  single-phase  generators  running  in  multiple  with 
the  general  three-phase  system,  and  controlling  voltage  and  me- 
chanical power  supply  so  as  to  absorb  the  single-phase  load  by  the 
single-phase  generators.     In  this  case,  however,  if  the  single- 
phase  load  uses  the  same  transmission  line  as  the  three-phase 
load,  phase  balancing  at  the  receiving  circuit  may  be  necessary. 

(c)  By    taking  the  single-phase  load   from  the   three-phase 
system.     If  the  load  is  considerable,  this  may  require  special 
construction  of  the  generators,  and  phase  balancers. 

(d)  By  taking  the  power  all  as  balanced  three-phase  power 
from  the  generating  system,  and  converting  the  required  amount 
to  single-phase,  by  phase  converter  and  phase  balancer.     This 
may  be  done  in  the  generating  station,  or  at  the  receiving  station 
where  the  single-phase  power  is  required. 

Assuming  that  in  addition  to  a  balanced  three-phase  load  of 
power,  PQ,  a  single-phase  load  of  power,  P,  is  required.  Estimating 
roughly,  that  the  single-phase  capacity  of  a  machine  structure  is 
half  the  three-phase  capacity  of  the  structure — which  probably 
is  not  far  wrong — then  the  use  of  single-phase  generators  gives 
us  Po-kw.  three-phase,  and  P-kw.  single-phase  generators,  and  as 
the  latter  is  equal  in  size  to  2  P-kw.  three-phase  capacity,  the 
total  machine  capacity  would  be  P0  +  2  P. 

Three-phase  generation  and  phase  conversion  would  require 
Po  +  P  kw.  in  three-phase  generators,  and  phase  converters 
transferring  half  the  single-phase  power  from  the  phase  which  is 
loaded  by  single-phase,  to  the  quadrature  phase.  That  is,  the 
phase  converter  must  have  a  capacity  of  P/2  kw.  in  the  motor 
phase,  and  P/2  kw.  capacity  in  the  generator  phase,  or  a  total 


PHASE  CONVERSION  233 

capacity  of  P  kw.  Thus  the  total  machine  capacity  required  for 
both  kinds  of  load  would  again  be  P0  +  2  P  kw.  three-phase 
rating. 

Thus,  as  regards  machine  capacity,  there  is  no  material  differ- 
ence between  single-phase  generation  and  three-phase  genera- 
tion with  phase  conversion,  and  the  decision  which  arrangement  is 
preferable  will  largely  depend  on  questions  of  construction  and 
operation.  A  more  complete  discussion  on  single-phase  genera- 
tion and  phase  conversion  is  given  in  A.  I.  E.  E.  Transactions, 
November,  1916. 


CHAPTER  XV 
SYNCHRONOUS  RECTIFIER 

SELF-COMPOUNDING    ALTERNATORS — SELF-STARTING    SYNCHRO- 
NOUS   MOTORS — ARC    RECTIFIER — BRUSH    AND    THOMSON 
HOUSTON  ARC  MACHINE — LEBLANC  PANCHAHUTEUR — 
PERMUTATOR — SYNCHRONOUS  CONVERTER 

138.  Rectifiers  for  converting  alternating  into  direct  current 
have  been  designed  and  built  since  many  years.  As  mechanical 
rectifiers,  mainly  single-phase,  they  have  found  a  limited  use  for 
small  powers  since  a  long  time,  and  during  the  last  years  arc 
rectifiers  have  found  extended  use  for  small  and  moderate  powers, 
for  storage-battery  charging  and  for  series  arc  lighting  by  constant 
direct  current.  For  large  powers,  however,  the  rectifier  does  not 
appear  applicable,  .but  the  synchronous  converter  takes  its  place. 
The  two  most  important  types  of  direct-current  arc-light  ma- 
chines, however,  have  in  reality  been  mechanical  rectifiers,  and 
for  compounding  alternators,  and  for  starting  synchronous 
motors,  rectifying  commutators  have  been  used  to  a  considerable 
extent. 

Let,  in  Fig.  72,  e  be  the  alternating  voltage  wave  of  the  supply 
source,  and  the  connections  of  the  receiver  circuit  with  this  sup- 
ply source  be  periodically  and  synchronously  reversed,  at  the 
zero  points  of  the  voltage  wave,  by  a  reversing  commutator 
driven  by  a  small  synchronous  motor,  shown  in  Fig.  73.  In  the 
receiver  circuit  the  voltage  wave  then  is  unidirectional  but  pul- 
sating, as  shown  by  eo  in  Fig.  74. 

If  receiver  circuit  and  supply  circuit  both  are  non-inductive, 
the  current  in  the  receiver  circuit  is  a  pulsating  unidirectional 
current,  shown  as  i0  in  dotted  lines  in  Fig.  74,  and  derived  from 
the  alternating  current,  i,  Fig.  72,  in  the  supply  circuit. 

If,  however,  the  receiver  circuit  is  inductive,  as  a  machine  field, 
then  the  current,  iQ,  in  Fig.  75,  pulsates  less  than  the  voltage,  e0, 
which  produces  it,  and  the  current  thus  does  not  go  down  to  zero, 
but  is  continuous,  and  its  pulsation  the  less,  the  higher  the  in- 
ductance. The  current,  i,  in  the  alternating  supply  circuit,  how- 

234 


SYNCHRONOUS  RECTIFIER 


235 


FIG.  72. — Alternating  sine  wave. 


AC  or 
DC 


FIG.  73. — Rectifying  commutator. 


FIG.  74. — Rectified  wave  on  non  inductive  load. 


FIG.  75. — Rectified  wave  on-inductive  load. 


FIG.  76. — Alternating  supply  wave  to  rectifier  on  inductive  load. 

• 


236 


ELECTRICAL  APPARATUS 


ever,  from  which  the  direct  current,  i0,  is  derived  by  reversal,  must 
go  through  zero  twice  during  each  period,  thus  must  have  the 
shape  shown  as  i  in  Fig.  76,  that  is,  must  abruptly  reverse.  If, 
however,  the  supply  circuit  contains  any  self-inductance — and 
every  circuit  contains  some  inductance — the  current  can  not 
change  instantly,  but  only  gradually,  the  slower,  the  higher  the 
inductance,  and  the  actual  current  in  the  supply  circuit  assumes 


FIG.  77. — Differential  current  on  rectifier  on  inductive  load. 

a  shape  like  that  shown  in  dotted  lines  in  Fig.  76.  Thus  the  cur- 
rent in  the  alternating  part  and  that  in  the  rectified  part  of  the 
circuit  can  not  be  the  same,  but  a  difference  must  exist,  as  shown 
as  i'  in  Fig.  77.  This  current,  i' ,  passes  between  the  two  parts 


AC 


FIG.  78. — Rectifier  with  A.C.  and  D.C.  shunt  resistance  for  inductive  load. 

of  the  circuit,  as  arc  at  the  rectifier  brushes,  and  causes  the  recti- 
fying commutator  to  spark,  if  there  is  any  appreciable  inductance 
in  the  circuit.  The  intensity  of  the  sparking  current  depends 
on  the  inductance  of  the  rectified  circuit,  its  duration  on  that  of 
the  alternating  supply  circuit. 

By  providing  a  byepath  for  this  differential  current,  i' ,  the 
sparking  is  mitigated,  and  thereby  the  amount  of  power,  which  can 
be  rectified,  increased.  This  is  done  by  shunting  a  non-inductive 
resistance  across  the  rectified  circuit,  r0,  or  across  the  alternating 
circuit,  r,  or  both,  as  shown  in  Fig.  78.  If  this  resistance  is  low, 
it  consumes  considerable  power  and  finally  increases  sparking 


SYNCHRONOUS  RECTIFIER  237 

by  the  increase  of  rectified  current ;  if  it  is  high,  it  has  little  effect. 
Furthermore,  this  resistance  should  vary  with  the  current. 

The  belt-driven  alternators  of  former  days  frequently  had  a 
compounding  series  field  excited  by  such  a  rectifying  commutator 
on  the  machine  shaft,  and  by  shunting  40  to  50  per  cent,  of  the 
power  through  the  two  resistance  shunts,  with  careful  setting  of 
brushes  as  much  as  2000  watts  have  been  rectified  from  single- 
phase  125-cycle  supply. 

Single-phase  synchronous  motors  were  started  by  such  recti- 
fying commutators  through  which  the  field  current  passed,  in 
series  with  the  armature,  and  the  first  long-distance  power  trans- 


>N 


o 


FIG.  79.  —  Open-circuit  rectifier.  FIG.  80.  —  Short-circuit  rectifier. 

mission  in  America  (Telluride)  was  originally  operated  with 
single-phase  machines  started  by  rectifying  commutator  —  the 
commutator,  however,  requiring  frequent  renewal. 

139.  The  reversal  of  connection  between  the  rectified  circuit 
and  the  supply  circuit  may  occur  either  over  open-circuit,  or 
over  short-circuit.  That  is,  either  the  rectified  circuit  is  first 
disconnected  from  the  supply  circuit  —  which  open-circuits  both 
—  and  then  connected  in  reverse  direction,  or  the  rectified  circuit 
is  connected  to  the  supply  circuit  in  reverse  direction,  before 
.being  disconnected  in  the  previous  direction  —  which  short-circuits 
both  circuits.  The  former,  open-circuit  rectification,  results  if 
the  width  of  the  gap  between  the  commutator  segments  is  greater 
than  the  width  of  the  brushes,  Fig.  79,  the  latter,  short-circuit 
rectification,  results  if  the  width  of  the  gap  is  less  than  the  width 
of  the  brushes,  Fig.  80. 

In  open-circuit  rectification,  the  alternating  and  the  rectified 
voltage  are  shown  as  e  and  e0  in  Fig.  81.  If  the  circuit  is  non- 
inductive,  the  rectified  current,  i0,  has  the  same  shape  as  the  vol- 


238 


ELECTRICAL  APPARATUS 


tage,  eo,  but  the  alternating  current,  i,  is  as  shown  in  Fig.  81  as  i. 
If  the  circuit  is  inductive,  vicious  sparking  occurs  in  this  case 
with  open-circuit  rectification,  as  the  brush  when  leaving  the 


FIG.  81. — Voltage  and  current  waves  in  open-circuit  rectifier  on  non-induc- 
tive load. 

commutator  segment  must  suddenly  interrupt  the  current.  That 
is,  the  current  does  not  stop  suddenly,  but  continues  to  flow  as 
an  arc  at  the  commutator  surface,  and  also,  when  making  con- 


FIG.  82. — Voltage  and  current  wave  in  open-circuit  rectifier  on  inductive 
load,  showing  sparking. 

tact  between  brush  and  segment,  the  current  does  not  instantly 
reach  full  value,  but  gradually,  and  the  current  wave  thus  is  as 
shown  as  i  and  iQ  in  Fig.  82,  where  the  shaded  area  is  the  arcing 
current  at  the  commutator. 
Sparkless  rectification  may  be  produced  in  a  circuit  of  moderate 


SYNCHRONOUS  RECITFIER 


239 


inductance,  with  open-circuit  rectification,  by  shifting  the  brushes 
so  that  the  brushes  open  the  circuit  only  at  the  moment  when 
the  (inductive)  current  has  reached  zero  value  or  nearly  so,  as 


M  N         .   N 

FIG.  83. — Voltage  waves  of  open-circuit  rectifier  with  shifted  brushes. 

shown  in  Figs.  83  and  84.  In  this  case,  the  brush  maintains  con- 
tact until  the  voltage,  e,  has  not  only  gone  to  zero,  but  reversed 
sufficiently  to  stop  the  current,  and  the  rectified  voltage  then  is 
shown  by  e0  in  Fig.  83,  the  current  by  i  and  iQ  in  Fig.  84. 


FIG.  84. — Current  waves  of  open-circuit  rectifier  with  shifted  brushes. 

140.  With  short-circuit  commutation  the  voltage  waves  are  as 
shown  by  e  and  eQ  in  Fig.  85.  With  a  non-inductive  supply  and 
non-inductive  receiving  circuit,  the  currents  would  be  as  shown 
by  i  and  IQ  in  Fig.  86.  That  is,  during  the  period  of  short-circuit, 


240 


ELECTRICAL  APPARATUS 


v~ 

\ 

\ 

/ 

/ 

FIG.  85. — Voltage'waves  of  short-circuit  rectifier. 


EIG.  86. — Current  waves  of  short-circuit  rectifier  on  non-inductive  load. 


FIG.    87. — Current  waves  of  short  circuit  rectifier  on  moderately  inductive 
load,  showing  flashing. 


SYNCHRONOUS  RECTIFIER 


241 


the  current  in  the  rectified  circuit  is  zero,  and  is  high,  is  the  short- 
circuit  current  of  the  supply  voltage,  in  the  supply  circuit. 

Inductance  in  the  rectified  circuit  retards  the  dying  out  of  the 
current,  but  also  retards  its  rise,  and  so  changes  the  rectified 
current  wave  to  the  shapes  shown — for  increasing  values  of  in- 
ductance— as  IQ  in  Figs.  87,  88  and  89. 


FIG.  88. — Current  waves  of  short-circuit  rectifier  on  inductive  load  at  the 

stability  limit. 

Inductance  in  the  supply  circuit  reduces  the  excess  current 
value  during  the  short-circuit  period,  and  finally  entirely  elimi- 
nates the  current  rise,  but  also  retards  the  decrease  and  reversal 
of  the  supply  current,  and  the  latter  thus  assumes  the  shapes 
shown — for  successively  increasing  values  of  inductance — as  i  in 
Figs.  87,  88  and  89. 


FIG.  89. — Current  waves  of  short-circuit  rectifier  on  highly  inductive  load, 
showing  sparking  but  no  flashing. 

As  seen,  in  Figs.  86  and  87,  the  alternating  supply  current  has 
during  the  short-circuit  reversed  and  reached  a  value  at  the  end 
of  the  short-circuit,  higher  than  the  rectified  current,  and  at  the 
moment  when  the  brush  leaves  the'  short-circuit,  a  considerable 
current  has  to  be  broken,  that  is,  sparking  occurs.  In  Figs.  86 
and  87,  this  differential  current  which  passes  as  arc  at  the  com- 
mutator, is  shown  by  the  dotted  area.  It  is  increasing  with  in- 

16 


242 


ELECTRICAL  APPARATUS 


creasing  spark  length,  that  is,  the  spark  or  arc  at  the  commutator 
has  no  tendency  to  go  out — except  if  the  inductance  is  very  small 
— but  persists :  flashing  around  the  commutator  occurs  and  short- 
circuits  the  supply  permanently. 


FIG.  90. — Voltage  wave  of    short-circuit  rectifier  with    shifted  brushes. 

In  Fig.  89,  the  alternating  current  at  the  end  of  the  short- 
circuit  has  not  yet  reversed,  and  a  considerable  differential 
current,  shown  by  the  dotted  area,  d,  passes  as  arc.  Vicious 


FIG.  91. — Current  waves  of  short-circuit  rectifier  with  inductive  load  and  the 
brushes  shifted  to  give  good  rectification. 

sparking  thus  occurs,  but  in  this  case  no  flashing  around  the 
commutator,  as  with  increasing  spark  length  the  differential 
current  decreases  and  finally  dies  out. 

In  Fig.  88,  the  alternating  current  at  the  end  of  the  short- 
circuit  has  just  reached  the  same  value  as  the  rectified  current, 


SYNCHRONOUS  RECTIFIER  243 

thus  no  current  change  and  no  sparking  occurs.  However,  if 
the  short-circuit  should  last  a  moment  longer,  a  rising  differential 
current  would  appear  and  cause  flashing  around  the  commutator. 
Thus,  Fig.  88  just  represents  the  stability  limit  between  the 
stable  (but  badly  sparking)  condition,  Fig.  89,  and  the  unstable 
or  flashing  conditions,  Figs.  87  and  86. 

By  shifting  the  brushes  so  as  to  establish  and  open  the  short- 
circuit  later,  as  shown  in  Fig.  90,  the  short-circuited  alternating 
e.m.f. — shown  dotted  in  Figs.  90  and  85 — ceases  to  be  symmet- 
rical, that  is,  averaging  zero  as  in  Fig.  85,  and  becomes  unsym- 
metrical,  with  an  average  of  the  same  sign  as  the  next  following 
voltage  wave.  It  thus  becomes  a  commutating  e.m.f.,  causes  a 
more  rapid  reversal  of  the  alternating  current  during  the  short- 
circuit  period,  and  the  circuit  conditions,  Fig.  89,  then  change  to 
that  of  Fig.  91.  That  is,  the  current  produced  by  the  short- 
circuited  alternating  voltage  has  at  the  end  of  the  short-circuit 
period  reached  nearly,  but  not  quite  the  same  value  as  the  recti- 
fied current,  and  a  short  faint  spark  occurs  due  to  the  differential 
current,  d.  This  Fig.  91  then  represents  about  the  best  condition 
of  stable,  and  practically  sparkless  commutation :  a  greater  brush 
shift  would  reach  the  stability  limit  similar  as  Fig.  88,  a  lesser 
brush  shift  leave  unnecessarily  severe  sparking,  as  Fig.  89. 

141.  Within  a  wide  range  of  current  and  of  inductance — espe- 
cially for  highly  inductive  circuits — practically  sparkless  and 
stable  rectification  can  be  secured  by  short-circuit  commutation 
by  varying  the  duration  of  the  short-circuit,  and  by  shifting  the 
brushes,  that  is,  changing  the  position  of  the  short-circuit  during 
the  voltage  cycle. 

Within  a  wide  range  of  current  and  of  inductance,  in  low-in- 
ductance circuits,  practically  sparkless  and  stable  rectification 
can  be  secured  also  by  open-circuit  rectification,  by  varying  the 
duration  of  the  open-circuit,  and  by  shifting  the  brushes. 

The  duration  of  open-circuit  or  short-circuit  can  be  varied  by 
the  use  of  two  brushes  in  parallel,  which  can  be  shifted  against 
each  other  so  as  to  span  a  lesser  or  greater  part  of  the  circumfer- 
ence of  the  commutator,  as  shown  in  Fig.  92. 

Short-circuit  commutation  is  more  applicable  to  circuits  of 
high,  open-circuit  commutation  to  circuits  of  low  inductance. 

But,  while  either  method  gives  good  rectification  if  overlap  and 
brush  shift  are  right,  they  require  a  shift  of  the  brushes  with  every 
change  of  load  or  of  inductivity  of  the  load,  and  this  limits  the 


244 


ELECTRICAL  APPARATUS 


practical  usefulness  of  rectification,  as  such  readjustment  with 
every  change  of  circuit  condition  is  hardly  practicable. 

Short-circuit  rectification  has  been  used  to  a  large  extent  on 
constant-current  circuits ;  it  is  the  method  by  which  the  Thomson- 


A 


o 


o 


V   M 


FIG.  92. — Double-brush  rectifier. 

Houston  (three-phase)  and  the  Brush  arc  machine  (quarter- 
phase)  commutates.  For  more  details  on  this  see  "  Theory  and 
Calculations  of  Transient  Phenomena,"  Section  II. 


FIG.  93. — Voltage  waves  of  open -circuit  rectifier  charging  storage  battery. 

Open-circuit  rectification  has  found  a  limited  use  on  non-in- 
ductive circuits  containing  a  counter  e.m.f.,  that  is,  in  charging 
storage  batteries. 

If,  in  Fig.  93,  eQ  is  the  rectified  voltage,  and  e±  the  counter  e.m.f. 

CQ  —  e 

of  the  storage  battery,  the  current  is  ^o  =  ~      ~'  where  r  =  ef- 
fective resistance  of  the  battery,  and  if  the  counter  e.m.f.  of  the 


SYNCHRONOUS  RECTIFIER 


245 


battery,  e\,  equals  the  initial  and  the  final  value  of  eQ)  as  in  Fig. 
93,  eo  —  .e  and  thus  i0  start  and  end  with  zero,  that  is,  no  abrupt 
change  of  current  occurs,  and  moderate  inductivity  thus  gives 
no  trouble.  The  current  waves  then  are:  i  and  i0  in  Fig.  94. 


FIG.  94. — Current  waves  of  open-circuit  rectifier  charging  storage  battery. 

142.  Rectifiers  may  be  divided  into  reversing  rectifiers,  like 
those  discussed  heretofore,  and  shown,  together  with  its  supply 
transformer,  in  Figs.  95  and  96,  and  contact-making  rectifiers, 
shown  in  Figs.  97  and  98,  or  in  its  simplest  form,  as  half- wave 
rectifier,  in  Fig.  99. 


HhhH 


FIG.  95. — Reversing  rectifier  with    FIG.    96. — Reversing     rectifier 
alternating-current  rotor.  with  direct-current  rotor. 

As  seen,  in  Fig.  99,  contact  is  made  between  the  rectified  cir- 
cuit and  the  alternating  supply  source,  T,  during  one-half  wave 
only,  but  the  circuit  is  open  during  the  reverse  half  wave,  and  the 
rectified  circuit,  B,  thus  carries  a  series  of  separate  impulses  of  cur- 
rent and  voltage  as  shown  in  Fig.  100  as  ii.  However,  in  this 
case  the  current  in  the  alternating  supply  circuit  is  unidirectional 
also,  is  the  same  current,  i\.  This  current  produces  in  the  trans- 
former, T,  a  unidirectional  magnetization,  and,  if  of  appreciable 


246 


ELECTRICAL  APPARATUS 


magnitude,  that  is,  larger  than  the  exciting  current  of  the  trans- 
former, it  saturates  the  transformer  iron.  Running  at  or,  beyond 
magnetic  saturation,  the  primary  exciting  current  of  the  trans- 
former then  becomes  excessive,  the  hysteresis  heating  due  to  the 
unsymmetrical  magnetic  cycle  is  greatly  increased,  and  the 
transformer  endangered  or  destroyed. 


FIG.  97. — Contact-making  rectifier 
with  direct-current  rotor. 


FIG.    98. — Contact-making   rectifier 
with  alternating-current  rotor. 


Half-wave  rectifiers  thus  are  impracticable  except  for  extremely 
small  power. 

The  full-wave  contact-making  rectifier,  Fig.  97  or  98,  does  not 
have  this  objection.  In  this  type  of  rectifier,  the  connection  be- 
tween rectified  receiver  circuit  and 
alternating  supply  circuit  are  not 
synchronously  reversed,  as  in  Fig.  95 
or  96,  but  in  Fig.  97  one  side  of  the 
rectified  circuit,  B,  is  permanently 
connected  to  the  middle  m  of  the 
alternating  supply  circuit,  T,  while  the 
other  side  of  the  rectified  circuit  is 
synchronously  connected  and  discon- 
nected with  the  two  sides,  a  and 
&,  of  the  alternating  supply  circuit. 
Or  we  may  say:  the  rectified  circuit  takes  one-half  wave  from 
the  one  transformer  half  coil,  ma,  the  other  half  wave  from 
the  other  transformer  half  coil,  mb.  Thus,  while  each  of  the  two 
transformer  half  coils  carries  unidirectional  current,  the  uni- 
directional currents  in  the  two  half  coils  flow  in  opposite  direc- 
tion, thus  give  magnetically  the  same  effect  as  one  alternating 


FIG.  99. — Half -wave  rectifier, 
contact  making. 


SYNCHRONOUS  RECTIFIER 


247 


current  in  one  half  coil,  and  no  unidirectional  magnetization  re- 
sults in  the  transformer. 

In  the  contact-making  rectifier,  Fig.  98,  the  two  halves  of  the 
rectified  circuit,  or  battery,  B,  alternately  receive  the  two  suc- 
cessive half  waves  of  the  transformer,  T. 

The  voltage  and  current  waves  of  the  rectifier,  Fig.  97,  are 
shown  in  Fig.  100.  e  is  the  voltage  wave  of  the  alternating  sup- 


FIG.  100.- 


-Voltage  and  current  waves  of  contact-making  rectifier  with 
direct-current  rotor. 


ply  source,  from  a  to  b.  e\  and  e2  then  are  the  voltage  waves  of 
the  two  half  coils,  am  and  bm,  ii  and  i%  the  two  currents  in  these 
two  half  coils,  and  IQ  the  rectified  current,  and  voltage  in  the 
circuit  from  ra  to  c.  The  current,  ii,  in  the  one,  and,  i2,  in  the  other 
half  coil,  naturally  has  magnetically  the  same  effect  on  the  pri- 
mary, as  the  current,  ii  +  i%  =  i0;m  one  half  coil,  or  the  current, 
io/2  =  i,  in  the  whole  coil,  ab,  would  have.  Thus  it  may  be  said : 
in  the  (full-wave)  contact-making  rectifier,  Fig.  97,  the  rectified 


248 


ELECTRICAL  APPARATUS 


voltage,  eo,  is  one-half  the  alternating  voltage,  e,  and  the  rectified 
current,  i0,  is  twice  the  alternating  current,  i.     However,  the  i2r 
in  the  secondary  coil,  ab,  is  greater,  by  \/2, 
than  it  would  be  with  the  alternating  cur- 
rent, i  =  io/2. 

Inversely,  in  the  contact-making  rectifier, 
Fig.  98,  the  rectified  voltage  is  twice  the 
alternating  voltage,  the  rectified  current 
half  the  alternating  current. 

Contact-making    rectifiers    of    the  type 
Fig.  97  are  extensively  used  as  arc  recti- 
fiers,   more   particularly    the   mercury-arc 
rectifier   shown   diagrammatically  in   Fig. 
FIG.  101. — Mercury-    101.     This   may    be    compared   with   Fig. 
arc  rectifier,     contact    97      That  is,  the  making  of  contact  during 
one  half  wave,  and  opening  it  during  the 

reverse    half    wave,    is    accomplished  not  by  mechanical  syn- 
chronous rotation,  but    by   the    use  of    the  arc   as   unidirec- 


HH-HH 


FIG.  102. — Diagram  of  mercury-arc  rectifier  with  its  reactances. 

tional    conductor:1   with   the 'voltage    gradient    in    one  direc- 
tion,   the    arc    conducts;    with    the    reverse  voltage  gradient 

1  See  Chapter  II  of  "Theory  and  Calculation  of  Electric  Circuits." 


SYNCHRONOUS  RECTIFIER 


249 


—the  other  half  wave — it  does  not  conduct.  A  large  induc- 
tance is  used  in  the  rectified  circuit,  to  reduce  the  pulsation  of 
current,  and  inductances  in  the  two  alternating  supply  circuits 
— either  separate  inductances,  or  the  internal  reactance  of  the 
transformer — to  prolong  and  thereby  overlap  the  two  half  waves, 
and  maintain  the  rectifying  mercury  arc  in  the  vacuum  tube.  A 
diagram  of  a  mercury-arc  rectifier  with  its  reactances,  Xi,  x2,  XQ, 


FIG.  103. — Voltage  and  current  waves  of  mercury-arc  rectifier. 


is  shown  in  Fig.  102.  The  "A.C.  reactances"  x\  and  x2  often 
are  a  part  of  the  supply  transformer;  the  "D.C.  reactance"  x0 
is  the  one  which  limits  the  pulsation  of  the  rectified  current.  The 
waves  of  currents,  i\,  i2  and  iQ,  as  overlapped  by  the  inductances, 
Zi,  x2  and  XQ,  are  shown  in  Fig.  103. 

Full  description  and  discussion  of  the  mercury-arc  rectifier  is 
contained  in  "Theory  and  Calculation  of  Transient  Phenomena," 
Section  II,  and  in  "Radiation,  Light  and  Illumination." 


250 


ELECTRICAL  APPARATUS 


143.  To  reduce  the  sparking  at  the  rectifying  commutator, 
the  gap  between  the  segments  may  be  divided  into  a  number  of 
gaps,  by  small  auxiliary  segments,  as  shown  in  Fig.  104,  and 
these  then  connected  to  intermediate  points  of  the  shunting  re- 


[if      '       ^1 


r' 


FIG.  104. — Rectifier  with  intermediate  segments. 

sistance,  r,  which  takes  the  differential  current,  io  —  i,  or  the 
auxiliary  segments  may  be  connected  to  intermediate  points  of 
the  winding  of  the  transformer,  T,  which  feeds  the  rectifier, 
through  resistances,  r',  and  the  supply  voltage  thus  successively 


FIG.  105. — Three-phase  F-connected  rectifier. 

rectified.  Or  both  arrangements  may  be  combined,  that  is,  the 
intermediate  segments  connected  to  intermediate  points  of  the 
resistance,  r,  and  intermediate  points  of  the  transformer  wind- 
ing, T. 

Polyphase  rectification  can  yield  somewhat  larger  power  than 


SYNCHRONOUS  RECTIFIER 


251 


FIG.  106. — Three-phase  7-connected    FIG.  107. — Three-phase  delta-con- 
rectifier,  simplified  diagram.  nected  rectifier. 


FIG.  108. — Quarter-phase  star-con- 
nected rectifier. 


FIG.  109.— Quarter-phase  rectifier 
with  independent  phases. 


FIG.  110.  —  Quarter- 
phase  ring-connected 
rectifier. 


FIG.    111.  —  Quarter-phase    rectifier  with  two 
commutators. 


252 


ELECTRICAL  APPARATUS 


single-phase  rectification.  In  polyphase  rectification,  the  seg- 
ments and  circuits  may  be  in  star  connection,  or  in  ring  connec- 
tion, or  independent. 

Thus,  Fig.  105  shows  the  arrangement  of  a  star-connected  (or 
F-connected)  three-phase  rectifier.  The  arrangement  of  Fig.  105 
is  shown  again  in  Fig.  106,  in  simpler  representation,  by  showing 
the  phases  of  the  alternating  supply  circuit,  and  their  relation 
to  each  other  and  to  the  rectifier  segments,  by  heavy  black  lines 
inside  of  the  commutator. 

Fig.  107  shows  a  ring  or  delta-connected  three-phase  rectifier. 

Fig.  108  a  star-connected  quarter-phase  rectifier  and  Fig. 
109  a  quarter-phase  rectifier  with  two  independent  quadra- 


FIG.  112. — Voltage  waves  of  quarter-phase  star-connected  rectifier. 

ture  phases,  while  Fig.  110  shows  a  ring-connected  quarter-phase 
rectifier. 

The  voltage  waves  of  the  two  coils  in  Fig.  109  are  shown  as 
ei  and  62  in  Fig.  112,  in  thin  lines,  and  the  rectified  voltage  by  the 
heavy  black  line,  e0,  in  Fig.  112.  As  seen,  in  star  connection,  the 
successive  phases  alternate  in  feeding  the  rectified  circuit,  but 
only  one  phase  is  in  circuit  at  a  time,  except  during  the  time  of 
the  overlap  of  the  brushes  when  passing  the  gap  between  suc- 
cessive segments.  At  that  time,  two  successive  phases  are  in 
multiple,  and  the  current  changes  from  the  phase  of  decreasing 
voltage  to  that  of  rising  voltage.  Only  a  part  of  the  voltage 
wave  is  thus  used.  The  unused  part  of  the  wave,  e\,  is  shown 
shaded  in  Fig.  112. 

Fig.  113  shows  the  voltages  of  the  four  phases,  ei,  e2,  63,  04,  in 
ring  connection,  Fig.  110,  .and  as  e0  the  rectified  voltage.  As 
seen,  in  this  case,  all  the  phases  are  always  in  circuit,  two  phases 
always  in  series,  except  during  the  overlap  of  the  brushes  at  the 
gap  between  the  segments,  when  a  phase  is  short-circuited  dur- 
ing commutation.  The  rectified  voltage  is  higher  than  that  of 
each  phase,  but  twice  as  many  coils  are  required  as  sources  of 
supply  voltage,  each  carrying  half  the  rectified  current. 


SYNCHRONOUS  RECTIFIER 


253 


By  using  two  commutators  in  series,  as  shown  in  Fig.  Ill,  the 
two  phases  can  be  retained  continuously  in  circuit  while  using 


FIG.  113. — Voltage  waves  of  water-phase  ring- connected  rectifier. 

only  two  coils — but  two  commutators  are  required.     The  voltage 
waves  then  are  shown  in  Fig.  114. 


FIG.  114.— Voltage  waves  of  quarter-phase  rectifier  with  two  commutators. 

A  star-connected  six-phase  rectifier  is  shown  in  Fig.  115,  with 
the  voltage  waves  in  Fig.  117.     The  unused  part  of  wave  e\  is 


FIG.  115. — Six-phase  star- 
connected  rectifier. 


FIG.  116. — Six-phase  ring- 
connected  rectifier. 


shown   shaded.     A   six-phase   ring-connected    rectifier   in    Fig. 
116,  with  the  voltage  waves  in  Fig.  118. 


254 


ELECTRICAL  APPARATUS 


144.  As  seen,  with  larger  number  of  phases,  star  connection 
becomes  less  and  less  economical,  as  a  lesser  part  of  the  alternat- 
ing voltage  wave  is  used  in  the  rectified  voltage :  in  quarter-phase 


FIG.  117. — Voltage  waves  of  six-phase  star-connected  rectifier. 

rectification  90°  or  one-half,  in  six-phase  rectification  60°  or 
one-third,  etc.     In  ring  connection,  however,  all  the  phases  are 


FIG.  118. — Voltage  waves  of  six-phase  ring-connected  rectifier. 

continuously  in  circuit,  and  thus  no  loss  of  economy  occurs  by 
the  use  of  the  higher  number  of  phases. 


FIG.  119. — Rectifying  commutators  of  the  Brush  arc  machine. 

Therefore,  ring  connection  is  generally  used  in  rectification 
of  a  larger  number  of  phases,  and  star  connection  is  never  used 
beyond  quarter-phase,  that  is,  four  phases,  and  where  a  higher 
number  of  phases  is  desired,  to  increase  the  output,  several 


SYNCHRONOUS  RECTIFIER  255 

rectifying  commutators  are  connected  in  series,  as  shown  in 
Fig.  119.  This  represents  two  quarter-phase  rectifiers  in  series 
displaced  from  each  other  by  45°,  that  is,  an  eight-phase  system. 
Three-phase  star-connected  rectification,  Fig.  106,  has  been 
used  in  the  Thomson-Houston  arc  machine,  and  quarter-phase 
rectification,  Fig.  108,  in  the  Brush  arc  machine,  and  for  larger 
powers,  several  such  commutators  were  connected  in  series,  as 
in  Fig.  119.  These  machines  are  polyphase  (constant-current) 


FIG.  120. — Counter  e.m.f.  shunting  gaps  of  six-phase  rectifier. 

alternators  connected  to  rectifying  commutators  on  the  armature 
shaft. 

For  a  more  complete  discussion  of  the  rectification  of  arc 
machine  see  "  Theory  and  Calculation  of  Transient  Electric 
Phenomena,"  Section  II. 

145.  Even  with  polyphase  rectification,  the  power  which  can 
be  rectified  is  greatly  limited  by  the  sparking  caused  by  the  dif- 
ferential current,  that  is,  the  difference  between  the  rectified 
current,  io,  which  never  reverses,  but  is  practically  constant,  and 
the  alternating  supply  current.  Resistances  shunting  the  gaps 
between  adjoining  segments,  as  byepath  for  this  differential  cur- 
rent, consume  power  and  mitigate  the  sparking  to  a  limited  extent 
only.  A  far  more  effective  method  of  eliminating  the  sparking 
is  by  shunting  this  differential  current  not  through  a  mere  non- 
inductive  resistance,  but  through  a  non-inductive  resistance  which 
contains  an  alternating  counter  e.m.f.  equal  to  that  of  the  supply 
phase,  as  shown  diagrammatically  in  Fig.  120. 

In  Fig.  120,  61  to  e&  are  the  six  phases  of  a  ring-connected  six- 
phase  system;  e'\  to  e's  are  e.m.fs.  of  very  low  self -inductance 


256 


ELECTRICAL  APPARATUS 


and  moderate  resistance,  r,  shunted  between  the  rectifier  seg- 
ments. Fig.  121  then  shows  the  wave  shape  of  the  current,  iQ  —  i, 
which  passes  through  these  counter  e.m.fs.,  ef  (assuming  that  the 
circuit  of  e',  r,  contains  no  appreciable  self -inductance) . 

Such  polyphase  counter  e.m.fs.  for  shunting  the  differential 
current  between  the  segments,  can  be  derived  from  the  syn- 
chronous motor  which  drives  the  rectifying  commutator.  By 
winding  the  synchronous-motor  armature  ring  connected  and 


FIG.  121. — Wave  shape  of  differential  current. 

of  the  same  number  of  phases  as  the  rectifying  commutator,  and 
using  a  revolving-armature  synchronous  motor,  the  synchronous- 
motor  armature  coils  can  be  connected  to  the  rectifier  segments, 
and  byepass  the  differential  current.  To  carry  this  current,  the 
armature  conductor  of  the  synchronous  motor  has  to  be  increased 
in  size,  but  as  the  differential  current  is  small,  this  is  relatively 


009696 


FIG.  122. — Leblanc's  Panchahuteur. 

little.  Hereby  the  output  which  can  be  derived  from  a  poly- 
phase rectifier  can  be  very  largely  increased,  the  more,  the  larger 
the  number  of  phases.  This  is  Leblanc's  Panchahuteur,  shown 
diagrammatically  in  Fig.  122  for  six  phases. 

Such  polyphase  rectifier  with  non-inductive  counter  e.m.f. 
byepath  through  the  synchronous-motor  armature  requires  as 
many  collector  rings  as  rectifier  segments.  It  can  rectify  large 
currents,  but  is  limited  in  the  voltage  per  phase,  that  is, 
per  segment,  to  20  to  30  volts  at  best,  and  the  larger  th 


SYNCHRONOUS  RECTIFIER 


257 


required  rectified  voltage,  the  larger  thus  must  be  the  number  of 
phases. 

146.  Any  number  of  phases  can  be  produced  in  the  secondary 
system  from  a  three-phase  or  quarter-phase  primary  polyphase 
system  by  transformation  through  two  or  three  suitably  designed 
stationary  transformers,  and  a  large  number  of  phases  thus  is 
not  objectionable  regarding  its  production  by  transformation. 
The  serious  objection  to  the  use  of  a  large  number  of  phases 
(24,  81,  etc.)  is,  that  each  phase  requires  a  collector  ring  to  lead 
the  current  to  the  corresponding  segment  of  the  rectifying 
commutator. 

This  objection  is  overcome  by  various  means: 

1.  The  rectifying  commutator  is  made  stationary  and  the 
brushes  revolving.  The  synchronous  motor  then  has  revolving 


FIG.  123. — Phase  splitting  by  synchronous-motor  armature:  synchronous 

converter. 

field  and  stationary  armature,  and  the  connection  from  the 
stationary  polyphase  transformer  to  the  commutator  segments 
and  the  armature  coils  is  by  stationary  leads. 

Such  a  machine  is  called  a  permutator.  It  has  been  built  to  a 
limited  extent  abroad.  It  offers  no  material  advantage  over  the 
synchronous  converter,  but  has  the  serious  disadvantage  of  re- 
volving brushes.  This  means,  that  the  brushes  can  not  be  in- 
spected or  adjusted  during  operation,  that  if  one  brush  sparks 
by  faulty  adjustment,  etc.,  it  is  practically  impossible  to  find  out 
which  brush  is  at  fault,  and  that  due  to  the  action  of  centrifugal 
forces  on  the  brushes,  the  liability  to  troubles  is  greatly  increased. 

17 


258  ELECTRICAL  APPARATUS 

For  this  reason,  the  permutator  has  never  been  introduced  in 
this  country,  and  has  practically  vanished  abroad. 

2.  The  transformer  is  mounted  on  the  revolving-motor  struc- 
ture,   thereby   revolving,    permitting   direct    connection   of   its 
secondary  leads  with  the  commutator  segments.     In  this  case 
only  the  three  or  four  primary  phases  have  to  be  lead  into  the 
rotor  by  collector  rings. 

The  mechanical  design  of  such  structure  is  difficult,  the  trans- 
former, not  open  to  inspection  during  operation,  and  exposed  to 
centrifugal  forces,  which  limit  its  design,  exclude  oil  and  thus 
limit  the  primary  voltage,  so  that  with  a  high-voltage  primary- 
supply  system,  double  transformation  becomes  necessary. 

As  this  construction  offers  no  material  advantage  over  (3), 
it  has  never  reached  beyond  experimental  design. 

3.  A  lesser  number  of  collector  rings  and  supply  phases  is 
used,  than  the  number  of  commutator  segments  and  synchronous- 
motor  armature  coils,  and  the  latter  are  used  as  autotransformers 
to  divide  each  supply  phase  into  two  or  more  phases  feeding  suc- 
cessive commutator  segments.     Fig.  123  shows  a  12-phase  recti- 
fying commutator  connected  to  a  12-phase  synchronous  motor 
with  six  collector  rings  for  a  six-phase  supply,  so  that  each  sup- 
ply phase  feeds  two  motor  phases  or  coils,  and  thereby  two  recti- 
fier segments.     Usually,  more  than  two  segments  are  used  per 
supply  phase.     The  larger  the  number  of  commutator  segments 
per  supply  phase,  the  larger  is  the  differential  current  in  the 
synchronous  motor  armature  coils,  and  the  larger  thus  must  be 
this  motor. 

Calculation,  however,  shows  that  there  is  practically  no  gain 
by  the  use  of  more  than  12  supply  phases,  and  very  little  gain 
beyond  six  supply  phases,  and  that  usually  the  most  economical 
design  is  that  using  six  supply  phases  and  collector  rings,  no 
matter  how  large  a  number  of  phases  is  used  on  the  commutator. 

Fig.  123  is  the  well-known  synchronous  converter,  which  hereby 
appears  as  the  final  development,  for  large  powers,  of  the  syn- 
chronous rectifier. 

This  is  the  reason  why  the  synchronous  rectifier  apparently 
has  never  been  developed  for  large  powers :  the  development  of  the 
polyphase  synchronous  rectifier  for  high  power,  by  increasing 
the  number  of  phases,  byepassing  the  differential  current  which 
causes  the  sparking,  by  shunting  the  commutator  segments  with 
the  armature  coils  of  the  motor,  and  finally  reducing  the  number 


SYNCHRONOUS  RECTIFIER  259 

of  collector  rings  and  supply  phases  by  phase  splitting  in  the 
synchronous-motor  armature,  leads  to  the  synchronous  con- 
verter as  the  final  development  of  the  high-power  polyphase 
rectifier. 

For  " synchronous  converter"  see  " Theoretical  Elements  of 
Electrical  Engineering/'  Part  II,  C.  For  some  special  types  of 
synchronous  converter  see  under  "Regulating  Pole  Converter" 
in  the  following  Chapter  XXI. 


CHAPTER  XVI 
REACTION  MACHINES 

147.  In  the  usual  treatment  of  synchronous  machines  and 
induction  machines,  the  assumption  is  made  that  the  reactance, 
x,  of  the  machine  is  a  constant.  While  this  is  more  or  less 
approximately  the  case  in  many  alternators,  in  others,  especially 
in  machines  of  large  armature  reaction,  the  reactance,  x,  is 
variable,  and  is  different  in  the  different  positions  of  the  armature 
coils  in  the  magnetic  circuit.  This  variation  of  the  reactance 
causes  phenomena  which  do  not  find  their  explanation  by  the 
theoretical  calculations  made  under  the  assumption  of  constant 
reactance. 

It  is  known  that  synchronous  motors  or  converters  of  large 
and  variable  reactance  keep  in  synchronism,  and  are  able  to  do 
a  considerable  amount  of  work,  and  even  carry  under  circum- 
stances full  load,  if  the  field-exciting  circuit  is  broken,  and  thereby 
the  counter  e.m.f.,  EI,  reduced  to  zero,  and  sometimes  even  if 
the  field  circuit  is  reversed  and  the  counter  e.m.f.,  EI,  made 
negative. 

Inversely,  under  certain  conditions  of  load,  the  current  and 
the  e.m.f,  of  a  generator  do  not  disappear  if  the  generator  field 
circuit  is  broken,  or  even  reversed  to  a  small  negative  value,  in 
which  latter  case  the  current  is  against  the  e.m.f.,  EQ,  of  the 
generator. 

Furthermore,  a  shuttle  armature  without  any  winding  (Fig. 
126)  will  in  an  alternating  magnetic  field  revolve  when  once 
brought  up  to  synchronism,  and  do  considerable  work  as  a  motor. 

These  phenomena  are  not  due  to  remanent  magnetism  nor 
to  the  magnetizing  effect  of  eddy  currents,  because  they  exist 
also  in  machines  with  laminated  fields,  and  exist  if  the  alternator 
is  brought  up  to  synchronism  by  external  means  and  the  rema- 
nent magnetism  of  the  field  poles  destroyed  beforehand  by 
application  of  an  alternating  current. 

These  phenomena  can  not  be  explained  under  the  assump- 
tion of  a  constant  synchronous  reactance;  because  in  this  case, 
at  no-field  excitation,  the  e.m.f.  or  counter  e.m.f.  of  the  machine 

260 


REACTION  MACHINES  261 

is  zero,  and  the  only  e.m.f.  existing  in  the  alternator  is  the  e.m.f. 
of  self-induction;  that  is,  the  e.m.f.  induced  by  the  alternating 
current  upon  itself.  If,  however,  the  synchronous  reactance  is 
constant,  the  counter  e.m.f.  of  self-induction  is  in  quadrature 
with  the  current  and  wattless;  that  is,  can  neither  produce  nor 
consume  energy. 

In  the  synchronous  motor  running  without  field  excitation, 
always  a  large  lag  of  the  current  behind  the  impressed  e.m.f. 
exists;  and  an  alternating-current  generator  will  yield  an  e.m.f. 
without  field  excitation  only  when  closed  by  an  external  circuit 
of  large  negative  reactance ;  that  is,  a  circuit  in  which  the  current 
leads  the  e.m.f.,  as  a  condenser,  or  an  overexcited  synchronous 
motor,  etc. 

148.  The  usual  explanation  of  the  operation  of  the  synchronous 
machine  without  field  excitation  is  self-excitation  by  reactive 
armature  currents.  In  a  synchronous  motor  a  lagging,  in  a 
generator  a  leading  armature  current  magnetizes  the  field,  and  in 
such  a  case,  even  without  any  direct-current  field  excitation,  there 
is  a  field  excitation  and  thus  a  magnetic  field  flux,  produced  by  the 
m.m.f.  of  the  reactive  component  of  the  armature  currents.  In 
the  polyphase  machine,  this  is  constant  in  intensity  and  direc- 
tion, in  the  single-phase  machine  constant  in  direction,  but  pul- 
sating in  intensity,  and  the  intensity  pulsation  can  be  reduced 
by  a  short-circuit  winding  around  the  field  structure,  as  more 
fully  discussed  under  ''Synchronous  Machines." 

Thus  a  machine  as  shown  diagrammatically  in  Fig.  124,  with 
a  polyphase  (three-phase)  current  impressed  on  the  rotating 
armature,  A,  and  no  winding  on  the  field  poles,  starts,  runs  up 
to  synchronous  and  does  considerable  work  as  synchronous 
motor,  and  under  load  may  even  give  a  fairly  good  (lagging)  power- 
factor.  With  a  single-phase  current  impressed  upon  the  arma- 
ture, A,  it  does  not  start,  but  when  brought  up  to  synchronism, 
continues  to  run  as  synchronous  motor.  Driven  by  mechanical 
power,  with  a  leading  current  load  it  is  a  generator. 

However,  the  operation  of  such  machines  depends  on  the 
existence  of  a  polar  field  structure,  that  is  a  structure  having  a 
low  reluctance  in  the  direction  of  the  field  poles,  P  —  P,  and  a 
high  reluctance  in  quadrature  position  thereto.  Or,  in  other 
words,  the  armature  reactance  with  the  coil  facing  the  field  poles 
is  high,  and  low  in  the  quadrature  position  thereto. 

In  a  structure  with  uniform  magnetic  reluctance,  in  which 


262  ELECTRICAL  APPARATUS 

therefore  the  armature  reactance  does  not  vary  with  the  posi- 
tion of  the  armature  in  the  field,  as  shown  in  Fig.  125,  such  self- 
excitation  by  reactive  armature  currents  does  not  occur,  and 
direct-current  field  excitation  is  always  necessary  (except  in  the 
so-called  " hysteresis  motor"). 

Vectorially  this  is  shown  in  Figs.  124  and  125  by  the  relative 
position  of  the  magnetic  flux,  4>,  the  voltage,  E,  in  quadrature  to 
<f>,  and  the  m.m.f.  of  the  current,  7.  In  Fig.  125,  where  I  and 
<I>  coincide,  7  and  E  are  in  quadrature,  that  is,  the  power  zero. 
Due  to  the  polar  structure  in  Fig.  124,  7  and  <£  do  not  coincide, 


FIG.  124. — Diagram  of  machine  with        FIG.  125. — Diagram  of  machine  with 
polar  structure.  uniform  reluctance. 

thus  7  is  not  in  quadrature  to  E,  but  contains  a  positive  or  a 
negative  energy  component,  making  the  machine  motor  or 
generator. 

As  the  voltage,  E,  is  produced  by  the  current,  7,  it  is  an  e.m.f . 
of  self-induction,  and  self -excitation  of  the  synchronous  machine 
by  armature  reaction  can  be  explained  by  the  fact  that  the 
counter  e.m.f.  of  self-induction  is  not  wattless  or  in  quadrature 
with  the  current,  but  contains  an  energy  component;  that  is, 
that  the  reactance  is  of  the  form  X  =  h  -f  jx,  where  x  is  the  watt- 
less component  of  reactance  and  h  the  energy  component  of 
reactance,  and  h  is  positive  if  the  reactance  consumes  power — 
in  which  case  the  counter  e.m.f.  of  self-induction  lags  more  than 
90°  behind  the  current — while  h  is  negative  if  the  reactance 
produces  power — in  which  case  the  counter  e.m.f.  of  self-induction 
lags  less  than  90°  behind  the  current. 

149.  A  case  of  this  nature  occurs  in  the  effect  of  hysteresis, 
from  a  different  point  of  view.  In  "Theory  and  Calcuation  of  Al- 
ternating Current"  it  was  shown,  that  magnetic  hysteresis  distorts 
the  current  wave  in  such  a  way  that  the  equivalent  sine  wave, 


REACTION  MACHINES  263 

that  is,  the  sine  wave  of  equal  effective  strength  and  equal  power 
with  the  distorted  wave,  is  in  advance  of  the  wave  of  magnetism 
by  what  is  called  the  angle  of  hysteretic  advance  of  phase  a. 
Since  the  e.m.f.  generated  by  the  magnetism,  or  counter  e.m.f. 
of  self-induction  lags  90°  behind  the  magnetism,  it  lags  90°  +  a 
behind  the  current;  that  is,  the  self-induction  in  a  circuit  contain- 
ing iron  is  not  in  quadrature  with  the  current  and  thereby 
wattless,  but  lags  more  than  90°  and  thereby  consumes  power,  so 
that  the  reactance  has  to  be  represented  by  X  =  h  +  jx,  where 
h  is  what  has  been  called  the  "effective  hysteretic  resistance." 

A  similar  phenomenon  takes  place  in  alternators  of  variable 
reactance,  or,  what  is  the  same,  variable  magnetic  reluctance. 

Operation  of  synchronous  machines  without  field  excitation 
is  most  conveniently  treated  by  resolving  the  synchronous 
reactance,  XQ,  in  its  two  components,  the  armature  reaction  and  the 
true  armature  reactance,  and  once  more  resolving  the  armature 
reaction  into  a  magnetizing  and  a  distorting  component,  and 
considering  only  the  former,  in  its  effect  on  the  field.  The  true 
armature  self-inductance  then  is  usually  assumed  as  constant. 
Or,  both  armature  reactance  and  self-inductance,  are  resolved 
into  the  two  quadrature  components,  in  line  and  in  quadrature 
with  the  field  poles,  as  shown  in  Chapters  XXI  and  XXIV  of 
''Alternating-Current  Phenomena,"  5th  edition. 

150.  However,  while  a  machine  comprising  a  stationary  single- 
phase  "field  coil,"  A,  and  a  shuttle-shaped  rotor,  R,  shown 
diagrammatically  as  bipolar  in  Fig.  126,  might  still  be  interpreted 
in  this  matter,  a  machine  as  shown  diagrammatically  in  Fig. 
127,  as  four-polar  machine,  hardly  allows  this  interpretation. 
In  Fig.  127,  during  each  complete  revolution  of  the  rotor,  R, 
it  four  times  closes  and  opens  the  magnetic  circuit  of  the  single- 
phase  alternating  coil,  A,  and  twice  during  the  revolution,  the 
magnetism  in  the  rotor,  R,  reverses. 

A  machine,  in  which  induction  takes  place  by  making  and 
breaking  (opening  and  closing)  of  the  magnetic  circuit,  or  in 
general,  by  the  periodic  variation  of  the  reluctance  of  the 
magnetic  circuit,  is  called  a  reaction  machine. 

Typical  forms  of  such  reaction  machines  are  shown  diagram- 
matically in  Figs.  126  and  127.  Fig.  126  is  a  bipolar,  Fig.  127 
is  a  four-polar  machine.  The  rotor  is  shown  in  the  position  of 
closed  magnetic  circuit,  but  the  position  of  open  magnetic  circuit 
is  shown  dotted. 


264 


ELECTRICAL  APPARATUS 


Instead  of  cutting  out  segments  of  the  rotor,  in  Fig.  126,  the 
same  effect  can  be  produced,  with  a  cylindrical  rotor,  by  a  short- 
circuited  turn,  S,  as  shown  in  Fig.  128,  This  gives  a  periodic 
variation  of  the  effective  reluctance,  from  a  minimum,  shown  in 
Fig.  128,  to  a  maximum  in  the  position  shown  in  dotted  lines  in 
Fig.  128. 

This  latter  structure  is  the  so-called  "  synchronous-induction 
motor,"  Chapter  VIII,  which  here  appears  as  a  special  form  of 
the  reaction  machine. 

If  a  direct  current  is  sent  through  the  winding  of  the  machine, 


FIG. 


126.— Bipolar  reac- 
tion machine. 


FIG.  127. — Four-polar      FIG.  128. — Synchronous- 
reaction  machine.      induction   motor  as  reac- 
tion machine. 


Fig.  126  or  127,  a  pulsating  voltage  and  current  is  produced  in 
this  winding.  By  having  two  separate  windings,  and  energizing 
the  one  by  a  direct  current,  we  get  a  converter,  from  direct  cur- 
rent in  the  first,  to  alternating  current  in  the  second  winding. 
The  maximum  voltage  in  the  second  winding  can  not  exceed  the 
voltage,  per  turn,  in  the  exciting  winding,  thus  is  very  limited, 
and  so  is  the  current.  Higher  values  are  secured  by  inserting  a 
high  inductance  in  series  in  the  direct-current  winding.  In  this 
case,  a  single  winding  may  be  used  and  the  alternating-circuit 
shunted  across  the  machine  terminals,  inside  of  the  inductance. 
151.  Obviously,  if  the  reactance  or  reluctance  is  variable,  it 
will  perform  a  complete  cycle  during  the  time  the  armature  coil 
moves  from  one  field  pole  to  the  next  field  pole,  that  is,  during 
one-half  wave  of  the  main  current.  That  is,  in  other  words, 
the  reluctance  and  reactance  vary  with  twice  the  frequency  of 
the  alternating  main  current.  Such  a  case  is  shown  in  Figs. 
129  and  130.  The  impressed  e.m.f.,  and  thus  at  negligible 
resistance,  the  counter  e.m.f.,  is  represented  by  the  sine  wave, 


REACTION  MACHINES 


265 


E,  thus  the  magnetism  produced  thereby  is  a  sine  wave,  <t>,  90° 
ahead  of  E.     The  reactance  is  represented  by  the  sine  wave,  x, 


FIG.  129. — Wave  shapes  in  reaction  machine  as  generator. 


\ 


\ 


\7 


A 


FIG.  130. — Wave  shape  in  reaction  machine  as  motor. 

varying  with  the  double  frequency  of  E,  and  shown  in  Fig.  129 
to  reach  the  maximum  value  during  the  rise  of  magnetism,  in 


f 


266 


ELECTRICAL  APPARATUS 


Fig.  130  during  the  decrease  of  magnetism.  The  current,  /, 
required  to  produce  the  magnetism,  3>,  is  found  from  <J>  and  x  in 
combination  with  the  cycle  of  molecular  magnetic  friction  of  the 
material,  and  the  power,  P,  is  the  product,  IE.  As  seen  in  Fig. 


- 


- 


^s 


FIG.  131. — Hysteresis  loop  of  reaction  machine  as  generator. 

129,  the  positive  part  of  P  is  larger  than  the  negative  part; 
that  is,  the  machine  produces  electrical  energy  as  generator. 
In  Fig.  130  the  negative  part  of  P  is  larger  than  the  positive; 


FIG.  132. — Hysteresis  loop  of  reaction  machine  as  motor. 

that  is,  the  machine  consumes  electrical  energy  and  produces 
mechanical  energy  as  synchronous  motor.  In  Figs.  131  and  132 
are  given  the  two  hysteretic  cycles  or  looped  curves,  $,  I  under 
the  two  conditions.  They  show  that,  due  to  the  variation  of 


REACTION  MACHINES  267 

reactance,  x,  in  the  first  case,  the  hysteretic  cycle  has  been  over- 
turned so  as  to  represent,  not  consumption,  but  production  of 
electrical  energy,  while  in  the  second  case  the  hysteretic  cycle  has 
been  widened,  representing  not  only  the  electrical  energy  consumed 
by  molecular  magnetic  friction,  but  also  the  mechanical  output. 

152.  It  is  evident  that  the  variation  of  reluctance  must  be 
symmetrical  with  regard  to  the  field  poles;  that  is,  that  the 
two  extreme  values  of  reluctance,  maximum  and  minimum,  will 
take  place  at  the  moment  when  the  armature  coil  stands  in  front 
of  the  field  pole,  and  at  the  moment  when  it  stands  midway 
between  the  field  poles. 

The  effect  of  this  periodic  variation  of  reluctance  is  a  distortion 
of  the  wave- of  e.m.f.,  or  of  the  wave  of  current,  or  of  both. 
Here  again,  as  before,  the  distorted  wave  can  be  replaced  by 
the  equivalent  sine  wave,  or  sine  wave  of  equal  effective  intensity 
and  equal  power. 

The  instantaneous  value  of  magnetism  produced  by  the 
armature  current — which  magnetism  generates  in  the  arma- 
ture conductor  the  e.m.f.  of  self-induction — is  proportional  to 
the  instantaneous  value  of  the  current  divided  by  the  instan- 
taneous value  of  the  reluctance.  Since  the  extreme  values  of 
the  reluctance  coincide  with  the  symmetrical  positions  of  the 
armature  with  regard  to  the  field  poles — that  is,  with  zero  and 
maximum  value  of  the  generated  e.m.f.,  J£o,  of  the  machine — 
it  follows  that,  if  the  current  is  in  phase  or  in  quadrature  with 
the  generated  e.m.f.,  EQ,  the  reluctance  wave  is  symmetrical  to 
the  current  wave,  and  the  wave  of  magnetism  therefore  sym- 
metrical to  the  current  wave  also.  Hence  the  equivalent  sine 
wave  of  magnetism  is  of  equal  phase  with  the  current  wave ;  that 
is,  the  e.m.f.  of  self-induction  lags  90°  behind  the  current,  or  is 
wattless. 

Thus  at  no-phase  displacement,  and  at  90°  phase  displace- 
ment, a  reaction  machine  can  neither  produce  electrical  power 
nor  mechanical  power. 

If,  however,  the  current  wave  differs  in  phase  from  the  wave 
of  e.m.f.  by  less  than  90°,  but  more  than  zero  degrees,  it  is  un- 
symmetrical  with  regard  to  the  reluctance  wave,  and  the  re- 
luctance will  be  higher  for  rising  current  than  for  decreasing  cur- 
rent, or  it  will  be  higher  for  decreasing  than  for  rising  current, 
according  to  the  phase  relation  of  current  with  regard  to  generated 
e.m.f.,  EQ. 


268  ELECTRICAL  APPARATUS 

In  the  first  case,  if  the  reluctance  is  higher  for  rising,  lower  for 
decreasing,  current,  the  magnetism,  which  is  proportional  to 
current  divided  by  reluctance,  is  higher  for  decreasing  than  for 
rising  current;  that  is,  its  equivalent  sine  wave  lags  behind  the 
sine  wave  of  current,  and  the  e.m.f.  or  self-induction  will  lag 
more  than  90°  behind  the  current;  that  is,  it  will  consume 
electrical  power,  and  thereby  deliver  mechanical  power,  and  do 
work  as  a  synchronous  motor. 

In  the  second  case,  if  the  reluctance  is  lower  for  rising,  and 
higher  for  decreasing,  current,  the  magnetism  is  higher  for  rising 
than  for  decreasing  current,  or  the  equivalent  sine  wave  of 
magnetism  leads  the  sine  wave  of  the  current,  and  the  counter 
e.m.f.  of  self-induction  lags  less  than  90°  behind. the  current; 
that  is,  yields  electric  power  as  generator,  and  thereby  consumes 
mechanical  power. 

In  the  first  case  the  reactance  will  be  represented  by  X  =  h  + 
jx,  as  in  the  case  of  hysteresis;  while  in  the  second  case  the 
reactance  will  be  represented  by  X  —  —  h  +  jx. 

153.  The  influence  of  the  periodical  variation  of  reactance 
will  obviously  depend  upon  the  nature  of  the  variation,  that  is, 
upon  the  shape  of  the  reactance  curve.  Since,  however,  no 
matter  what  shape  the  wave  has,  it  can  always  be  resolved  in  a 
series  of  sine  waves  of  double  frequency,  and  its  higher  har- 
monics, in  first  approximation  the  assumption  can  be  made 
that  the  reactance  or  the  reluctance  varies  with  double  frequency 
of  the  main  current ;  that  is,  is  represented  in  the  form : 

x  =  a  +  &  cos  2  ]8. 
Let  the  inductance  be  represented  by: 

L  =  I  +  V  cos  2  ft 
=  1(1  +  7  cos  2  /3); 

where  7  =  amplitude  of  variation  of  inductance. 
Let: 

6  =  angle  of  lag  of  zero  value  of  current  behind  maximum  value 
of  the  inductance,  L. 

Then,  assuming  the  current  as  sine  wave,  or  replacing  it  by 
the  equivalent  sine  wave  of  effective  intensity,  7,  current: 

i  =  I  A/2  sin  (0  -  0). 


REACTION  MACHINES  269 


The  magnetism  produced  by  this  current  is: 

$=  — > 

n 

where  n  =  number  of  turns. 
Hence,  substituted: 


n  (0  _  0)  (i  +  y  Cos  2  0), 
n 

or,  expanded: 


-      cos  *  sn  0  "    1  +      sin  *  cos 


when  neglecting  the  term  of  triple  frequency  as  wattless. 
Thus  the  e.m.f.  generated  by  this  magnetism  is: 


e=    ~ 


hence,  expanded: 

e  =  -2  irfll  V2  |  (l  -  I)  cos  e  cos  0  +  (l  +  |)  sin  0  sin 
and  the  effective  value  of  e.m.f.  : 


E  =  27T/77       {l  -|)2cos2^+  (l  +|)2sin2  0 


=  2-irflI  ^l  +    -  -  7  cos  2  6. 
Hence,  the  apparent  power,  or  the  volt-amperes: 
Q  =  IE  =  2irflpJl  +^--7cos20 


-  7  cos  2  (9 

The  instantaneous  value  of  power  is:    . 

p  =  ei 

=  -47T/7/2  sin  (0  -  0)  [  (l  -  I)  cos  61  cos  0  + 

in  0  sin  0}; 


270  ELECTRICAL  APPARATUS 

and, expanded: 

p  =  -27T/7/2  !  (l  +  1}  sin  2  0  sin2  0  -  (l  -  ^ 

I  \         .J/  \         ^ 

sin  2  0  cos2  j3  +  sin  2  0  (cos  2  6  -  |)  1  • 

Integrated,  the  effective  value  of  power  is: 
p  =  -7r/7/27sin20; 

hence,  negative,  that  is,  the  machine  consumes  electrical,  and 
produces  mechanical,  power,  as  synchronous  motor,  if  6  >  0, 
that  is,  with  lagging  current;  positive,  that  is,  the  machine  pro- 
duces electrical,  and  consumes  mechanical  power,  as  generator, 
if  6  >  0,  that  is,  with  leading  current. 
The  power-factor  is : 

P  7  sin  2  6 

P=:~Q        ~~T  ^^ 

2  A/1  +  ~  -  7  cos  2  6 
hence,  a  maximum,  if: 

de  =  0; 

or,  expanded: 

cos  20  =  —  and  =  ~ 
7  2 

The  power,  P,  is  a  maximum  at  given  current,  /,  if: 

sin  2  0  =  1 ; 
that  is : 

0  =  45°; 

at  given  e.m.f.,  E,  the  power  is: 

p  = ff  27  sin  2  0 

47r/7(l  +^--7  cos  2 
hence,  a  maximum  at : 

dB  =  °; 
or,  expanded: 

±7 


cos  2  0  = 


-v2 


REACTION  MACHINES  271 

154.  We  have  thus,  at  impressed  e.m.f.,  E}  and  negligible 
resistance,  if  we  denote  the  mean  value  of  reactance: 

x  =  2  7T/7. 
Current: 

7  =  E 


X 

Volt-amperes : 


3  A/ 


x  1     •  -r  •  -  7  cos 

,  Power: 

#27  sin  2 

P  =  — 


+  -A 7  cos  2  0) 

4  / 

Power-factor : 

f-v   T\  7  sin  2  0 

p  =  cos  (E,  I)  = 


/ v*~ 

2  Jl  +  ^  -  7  cos  20 

Maximum  power  at: 

cos  2  0  =  — ^-v 


Maximum  power-factor  at: 


2  7 

cos  2  0  =  —  and   =  ^-- 


0  >  0:  synchronous  motor,  with  lagging  current, 
0  <  0 :  generator,  with  leading  current. 

As  an  example  is  shown  in  Fig.  133,  with  angle  0  as  abscissae, 
the  values  of  current,  power,  and  power-factor,  for  the  constants, 

T?    _    lift     v    =    2     Qnrl  >v    =    08 


E  =  110,  x  =  3,  and  7  =  0.8. 


I  =  -       _41 

"  Vl.45  -  cos  2  0' 
=    -2017  sin  2  0 
=  1.45  -  cos  2  0' 

/ET  0.447  sin  2  0 

p  =  cos  (#,  /)  =      , 

-  Vl.45  -  cos  2  0 


272 


ELECTRICAL  APPARATUS 


As  seen  from  Fig.  133,  the  power-factor,  p,  of  such  a  machine 
is  very  low — does  not  exceed  40  per  cent,  in  this  instance. 

Very  similar  to  the  reaction  machine  in  principle  and  character 
of  operation  are  the  synchronous  induction  motor,  Chapter  IX, 
and  the  hysteresis  motor,  Chapter  X,  either  of  which  is  a  gen- 
erator above  synchronism,  and  at  synchronism  can  be  motor  as 


FIG.  133. — Load  curves  of  reaction  machine. 

well  as  generator,  depending  on  the  relative  position  between 
stator  field  and  rotor. 

155.  The  low  power-factor  and  the  low  weight  efficiency  bar 
the  reaction  machine  from  extended  use  for  large  powers.  So 
also  does  the  severe  wave-shape  distortion  produced  by  it,  and 
it  thus  has  found  a  very  limited  use  only  in  small  sizes. 

It  has,  however,  the  advantage  of  a  high  degree  of  exactness 
in  keeping  in  step,  that  is,  it  does  not  merely  keep  in  synchronism 
and  drifts  more  or  less  over  a  phase  angle  with  respect  to  the 


REACTION  MACHINES  273 

impressed  voltage,  but  the  relative  position  of  the  rotor  with 
regards  to  the  phase  of  the  impressed  voltage  is  more  accurately 
maintained.  Where  this  feature  is  of  importance,  as  in  driving 
a  contact-maker,  a  phase  indicator  or  a  rectifying  commutator, 
the  reaction  machine  has  an  advantage,  especially  in  a  system 
of  fluctuating  frequency,  and  it  is  used  to  some  extent  for  such 
purposes. 

This  feature  of  exact  step  relation  is  shared  also,  though  to 
a  lesser  extent,  by  the  synchronous  motor  with  self-excitation 
by  lagging  currents,  and  ordinarily  small  synchronous  motors, 
but  without  field  excitation  (or  with  great  underexcitation  or 
overexcitation)  are  often  used  for  the  same  purpose. 

Machines  having  more  or  less  the  characteristics  of  the  reac- 
tion machine  have  been  used  to  a  considerable  extent  in  the 
very  early  days,  for  generating  constant  alternating  current  for 
series  arc  lighting  by  Jablochkoff  candles,  in  the  70's  and  early 
8Q's. 

Structurally,  the  reaction  machine  is  similar  to  the  inductor 
machine,  but  the  essential  difference  is,  that  the  former  operates 
by  making  and  breaking  the  magnetic  circuit,  that  is,  periodically 
changing  the  magnetic  flux,  while  the  inductor  machine  operates 
by  commutating  the  magnetic  flux,  that  is,  periodically  changing 
the  flux  path,  but  without  varying  the  total  value  of  the  magnetic 
flux. 


18 


CHAPTER  XVII 

INDUCTOR  MACHINES 

INDUCTOR  ALTERNATORS,  ETC. 

156.  Synchronous  machines  may  be  built  with  stationary 
field  and  revolving  armature,  as  shown  diagrammatically  in 
Fig.  134,  or  with  revolving  field  and  stationary  armature,  Fig. 
135,  or  with  stationary  field  and  stationary  armature,  but 
revolving  magnetic  circuit. 

The  revolving-armature  type  was  the  most  frequent  in  the 
early  days,  but  has  practically  gone  out  of  use  except  for  special 


FIG.  134. — Revolving  armature 
alternator 


FIG.     135. — Revolving  field   al- 
ternator. 


purposes,  and  for  synchronous  commutating  machines,  as  the 
revolving-armature  type  of  structure  is  almost  exclusively  used 
for  commutating  machines.  The  revolving-field  type  is  now 
almost  exclusively  used,  as  the  standard  construction  of  alter- 
nators, synchronous  motors,  etc.  The  inductor  type  had  been 
used  to  a  considerable  extent,  and  had  a  high  reputation  in  the 
Stanley  alternator.  It  has  practically  gone  out  of  use  for 
standard  frequencies,  due  to  its  lower  economy  in  the  use  of 
materials,  but  has  remained  a  very  important  type  of  construc- 
tion, as  it  is  especially  adapted  for  high  frequencies  and  other 
special  conditions,  and  in  this  field,  its  use  is  rapidly  increasing. 
A  typical  inductor  alternator  is  shown  in  Fig.  136,  as  eight- 
polar  quarter-phase  machine. 

274 


INDUCTOR  MACHINES 


275 


Its  armature  coils,  A,  are  stationary.  One  stationary  field 
coil,  F,  surrounds  the  magnetic  circuit  of  the  machine,  which 
consists  of  two  sections,  the  stationary  external  one,  B,  which 
contains  the  armature,  A,  and  a  movable  one,  C,  which  contains 
the  inductor,  N.  The  inductor  contains  as  many  polar  projec- 
tions, N,  as  there  are  cycles  or  pairs  of  poles.  The  magnetic  flux 
in  the  air  gap  and  inductor  does  not  reverse  or  alternate,  as  in 
the  revolving-field  type  of  alternator,  Fig.  135,  but  is  constant 
in  direction,  that  is,  all  the  inductor  teeth  are  of  the  same 
polarity,  but  the  flux  density  varies  or  pulsates,  between  a  maxi- 
mum, 'Bij  in  front  of  the  inductor  teeth,  and  a  minimum,  B2, 
though  in  the  same  direction,  in  front  of  the  inductor  slots.  The 
magnetic  flux,  3>,  which  interlinks  with  the  armature  coils,  does 
not  alternate  between  two  equal  and  opposite  values,  +  $o  and 


FIG.  136. — Inductor  alternator. 

—  $o,  as  in  Fig.  135,  but  pulsates  between  a  high  value,  $1, 
when  an  inductor  tooth  stands  in  front  of  the  armature  coil, 
and  a  low  value  in  the  same  direction,  <J>2,  when  the  armature 
coil  faces  an  inductor  slot. 

157.  In  the  inductor  alternator,  the  voltage  induction  thus 
is  brought  about  by  shifting  the  magnetic  flux  produced  by  a 
stationary  field  coil,  or  by  what  may  be  called  magneto  commu- 
tation, by  means  of  the  inductor. 

The  flux  variation,  which  induces  the  voltage  in  the  armature 
turns  of  the  inductor  alternator,  thus  is  $1  —  <f>2,  while  that  in 
the  revolving-field  or  revolving-armature  type  of  alternator  is 
2  <f>0. 

The  general  formula  of  voltage  induction  in  an  alternator  is: 

e  =  \/2 


276  ELECTRICAL  APPARATUS 

where  : 

/  =  frequency,  in  hundreds  of  cycles, 
n  =  number  of  armature  turns  in  series, 
$>o  =  maximum    magnetic    flux,    alternating 
through  the  armature  turns,  in  megalines, 
e  =  effective  value  of  induced  voltage. 

$1  —  $2  taking  the  place  of  2  3>0,  in  the  inductor  alternator, 
the  equation  of  voltage  induction  thus  is: 

n*1  ~  *'•  (2) 


As  seen,  3>i  must  be  more  than  twice  as  large  as  $0,  that  is, 
in  an  inductor  alternator,  the  maximum  magnetic  flux  interlinked 
with  the  armature  coil  must  be  more  than  twice  as  large  as  in  the 
standard  type  of  alternator. 

In  modern  machine  design,  with  the  efficient  methods  of  cool- 
ing now  available,  economy  of  materials  and  usually  also  effi- 
ciency make  it  necessary  to  run  the  flux  density  up  to  near  satura- 
tion at  the  narrowest  part  of  the  magnetic  circuit  —  which  usually 
is  the  armature  tooth.  Thus  the  flux,  <I>o,  is  limited  merely  by 
magnetic  saturation,  and  in  .the  inductor  alternator,  $1,  would  be 
limited  to  nearly  the  same  value  as,  $0,  in  the  standard  machine, 

(J)  ..       _      ^^O 

and  --  ~  --  thus  would  be  only  about  one-half  or  less  of  the 

permissible  value  of  3>o.  That  is,  the  output  of  the  inductor 
alternator  armature  is  only  about  one-half  that  of  the  standard 
alternator  armature.  This  is  obvious,  as  we  would  double  the 
voltage  of  the  inductor  alternator  armature,  if  instead  of  pulsat- 
ing between  <f>i  and  3>2  or  approximately  zero,  we  would  alternate 
between  $1  and  —  $1. 

On  the  other  hand,  the  single  field-coil  construction  gives  a 
material  advantage  in  the  material  economy  of  the  field,  and 
in  machines  having  very  many  field  poles,  that  is,  high-frequency 
alternators,  the  economy  in  the  field  construction  overbalances 
the  lesser  economy  in  the  use  of  the  armature,  especially  as  at 
high  frequencies  it  is  not  feasible  any  more  to  push  the  alter- 
nating flux,  $o,  up  to  or  near  saturation  values.  Therefore,  for 
high-frequency  generators,  the  inductor  alternator  becomes 
the  economically  superior  types,  and  is  preferred,  and  for  ex- 
tremely high  frequencies  (20,000  to  100,000  cycles)  the  inductor 
alternator  becomes  the  only  feasible  type,  mechanically. 

158.  In  the  calculation  of  the  magnetic  circuit  of  the  inductor 


INDUCTOR  MACHINES  277 

alternator,  if  3>0  is  the  amplitude  of  flux  pulsation  through  the 
armature  coil,  as  derived  from  the  required  induced  voltage  by 
equation  (1),  let: 

p  =  number  of  inductor  teeth,  that  is, 
number  of  pairs  of  poles  (four  in 
the  eight-polar  machine,  Fig.  136). 
Pi  =  magnetic  reluctance  of  air  gap  in  front 
of  the  inductor  tooth,  which  should 
be  as  low  as  possible, 

p2  =  magnetic  reluctance  of  leakage  path 
through  inductor  slot  into  the  arma- 
ture coil,  which  should  be  as  high  as 
possible, 

it  is:  11.  /Q\ 

$1  -f-  $2  =  —  •*•  — ;  (3) 

Pi  P2 

and  as: 

^^i     ~~~     4^o    rr    2    (&n  (4) 

it  follows:  P2 


=   2 


P2  —  PI 
Pi 


P2  -  PI 

and  the  total  flux  through  the  magnetic  circuit,  C,  and  out  from 
all  the  p  inductor  teeth  and  slots  thus  is : 

^fr      ~T"      *T1      (    ^fr-i      — [—      ^^O  ) 

P2  +  Pi 


P2    —    Pi 

)  «.         1 

(6) 


=  2 


P2    —    Pi 

In  the  corresponding  standard  alternator,  with  2  p  poles,  the 
total  flux  entering  the  armature  is : 

2p$0 

and  if  pi  is  the  reluctance  of  the  air  gap  between  field  pole  and 
armature  face,  p2  the  leakage  reluctance  between  the  field  poles, 
the  ratio  of  the  leakage  flux  between  the  field  poles,  <£',  to  the 
armature  flux,  3>0,  is: 

$0-^=1-1;  (7) 

Pi  P2 

hence: 

*'  =  *„->  (8) 

P2 


278 


ELECTRICAL  APPARATUS 


and  the  flux  in  the  field  pole,  thus,  is  : 


P2 

hence  the  total  magnetic  flux  of  the  machine,  of  2  p  poles: 

2pi\ 

P2 


'  (9) 


As  in  (6),  pi  is  small  compared  with  p2, 


2Pi 


P2    —    Pi 


.in  (6)  differs 


Pi 


little  from  -  -  in  (9).     That  is: 

P2 

As  regards  to  the  total  magnetic  flux  required  for  the  induc- 
tion of  the  same  voltage  in  the  same  armature,  no  material 
difference  exists  between  the  inductor  machine  and  the  standard 
machine ;  but  in  the  armature  teeth  the  inductor  machine  requires 
more  than  twice  the  maximum  magnetic  flux  of  the  standard 


FIG.  137. — Stanley  inductor  alternator. 

alternator,  and  thereby  is  at  a  disadvantage  where  the  limit 
of  magnetic  density  in  the  armature  is  set  only  by  magnetic 
saturation. 

As  regards  to  the  hysteresis  loss  in  the  armature  of  the  in- 
ductor alternator,  the  magnetic  cycle  is  an  unsymmetrical  cycle, 
between  two  values  of  the  same  direction,  BI  and  B2,  and  the 
loss  therefore  is  materially  greater  than  it  would  be  with  a 
symmetrical  cycle  of  the  same  amplitude.  It  is  given  by: 


where : 


770 


77  [1  +  0  B»]. 


INDUCTOR  MACHINES 


279 


Regarding  hereto  see  "  Theory  and  Calculation  of  Electric 
Circuits,"  under  "  Magnetic  Constants." 

However,  as  by  the  saturation  limit,  the  amplitude  of  the 
magnetic  pulsation  in  the  inductor  machine  may  have  to  be 
kept  very  much  lower  than  in  the  standard  type,  the  core  loss 
of  the  machine  may  be  no  larger,  or  may  even  be  smaller  than 
that  of  the  standard  type,  in  spite  of  the  higher  hysteresis 
coefficient,  170. 

159.  The  inductor-machine   type,   Fig.    136,   must  have   an 


FIG.  138. — Alexanderson  high  frequency  inductor  alternator. 

auxiliary  air  gap  in  the  magnetic  circuit,  separating  the  revolving 
from  the  stationary  part,  as  shown  at  S. 

It,  therefore,  is  preferable  10  double  the  structure,  Fig.  136, 
by  using  two  armatures  and  inductors,  with  the  field  coil  between 
them,  as  shown  in  Fig.  137.  This  type  of  alternator  has  been 
extensively  built,  as  the  Stanley  alternator,  mainly  for  60  cycles, 
and  has  been  a  very  good  and  successful  machine,  but  has  been 
superseded  by  the  revolving-field  type,  due  to  the  smaller  size 
and  cost  of  the  latter. 

Fig.  137  shows  the  magnetic  return  circuit,  B,  between  the  two 
armatures,  A,  and  the  two  inductors  N  and  S  as  constructed  of  a 
number  of  large  wrought-iron  bolts,  while  Fig.  136  shows  the 
return  as  a  solid  cast  shell. 


280 


ELECTRICAL  APPARATUS 


A  modification  of  this  type  of  inductor  machine  is  the  Alex- 
anderson  inductor  alternator,  shown  in  Fig.  138,  which  is  being 
built  for  frequencies  up  to  200,000  cycles  per  second  and  over, 
for  use  in  wireless  telegraphy  and  telephony. 

The  inductor  disc,  7,  contains  many  hundred  inductor  teeth, 
.  and  revolves  at  many  thousands  of  revolutions  between  the 
two  armatures,  A,  as  shown  in  the  enlarged  section,  S.  It  is 
surrounded  by  the  field  coil,  F,  and  outside  thereof  the  magnetic 
return,  B.  The  armature  winding  is  a  single-turn  wave  winding 
threaded  through  the  armature  faces,  as  shown  in  section  S  and 
face  view,  Q.  It  is  obvious  that  in  the  armature  special  iron 
of  extreme  thinness  of  lamination  has  to  be  used,  and  the  rotat- 
ing inductor,  /,  built  to  stand  the  enormous  centrifugal  stresses 
of  the  great  peripheral  speed.  We  must  realize  that  even  with 

an  armature  pitch  of  less  than  Ko  m- 
per  pole,  we  get  at  100,000  cycles  per 
second  peripheral  speeds  approaching 
bullet  velocities,  over  1000  miles  per 
hour.  For  the  lower  frequencies  of 
long  distance  radio  communication, 
20,000  to  30,000  cycles,  such  ma- 
chines have  been  built  for  large 
powers. 

160.  Fig.  139  shows  the  Eicke- 
meyer  type  of  inductor  alternator. 
In  this,  the  field  coil  F  is  not  con- 
centric to  the  shaft,  and  the  inductor 
teeth  not  all  of  the  same  polarity,  but 
the  field  coil,  as  seen  in  Fig.  139,  sur- 
rounds the  inductor,  I,  longitudinally, 

and  with  the  magnetic  return  B  thus  gives  a  bipolar  magnetic 
field.  Half  the  inductor  teeth,  the  one  side  of  the  inductor,  thus 
are  of  the  one,  the  other  half  of  the  other  polarity,  and  the 
armature  coils,  A,  are  located  in  the  (laminated)  pole  faces  of  the 
bipolar  magnetic  structure.  Obviously,  in  larger  machines,  a 
multipolar  structure  could  be  used  instead  of  the  bipolar  of  Fig. 
139.  This  type  has  the  advantage  of  a  simpler  magnetic  struc- 
ture, and  the  further  advantage,  that  all  the  magnetic  flux 
passes  at  right  angles  to  the  shaft,  just  as  in  the  revolving  field 
or  revolving  armature  alternator.  In  the  types,  Figs.  136  and 
137,  magnetic  flux  passes,  and  the  field  exciting  coil  magnetizes 


FIG.    139. — Eickemeyer    in- 
ductor alternator. 


INDUCTOR  MACHINES  281 

longitudinally  to  the  shaft,  and  thus  magnetic  stray  flux  tends 
to  pass  along  the  shaft,  closing  through  bearings  and  supports, 
and  causing  heating  of  bearings.  Therefore,  in  the  types  136 
and  137,  magnetic  barrier  coils  have  been  used  where  needed, 
that  is,  coils  concentric  to  the  shaft,  that  is,  parallel  to  the  field 
coil,  and  outside  of  the  inductor,  that  is,  between  inductor  and 
bearings,  energized  in  opposite  direction  to  the  field  coils.  These 
coils  then  act  as  counter-magnetizing  coils  in  keeping  magnetic 
flux  out  of  the  machine  bearings. 

The  type,  Fig.  139,  is  especially  adapted  for  moderate  fre- 
quencies, a  few  hundreds  to  thousands  of  cycles.  A  modifica- 
tion of  it,  adopted  as  converter,  is  used  to  a  considerable  extent : 
the  inductor,  7,  is  supplied  with  a  bipolar  winding  connected  to  a 
commutator,  and  the  machine  therefore  is  a  bipolar  commutating 
machine  in  addition  to  a  high-frequency  inductor  alternator 
(16-polar  in  Fig.  139).  It  thus  may  be  operated  as  converter, 
receiving  power  by  direct-current  supply,  as  direct-current  motor, 
and  producing  high-frequency  alternating  power  in  the  inductor 
pole-face  winding. 

161.  If  the  inductor  alternator,  Fig.  139,  instead  of  with  direct 
current,  is  excited  with  low-frequency  alternating  current,  that 


FIG.  140. — Voltage  wave  of  inductor  alternator  with  single-phase  excitation. 

is,  an  alternating  current  passed  through  the  field  coil,  F,  of  a 
frequency  low  compared  with  that  generated  by  the  machine  as 
inductor  alternator,  then  the  high-frequency  current  generated 
by  the  machine  as  inductor  alternator  is  not  of  constant  ampli- 
tude, but  of  a  periodically  varying  amplitude,  as  shown  in  Fig. 
140.  For  instance,  with  60-cycle  excitation,  a  64-polar  in- 
ductor (that  is,  inductor  with  32  teeth),  and  a  speed  of  1800 
revolutions,  we  get  a  frequency  of  approximately  1000  cycles, 
and  a  voltage  and  current  wave  about  as  shown  in  Fig.  140. 

The  power  required  for  excitation  obviously  is  small  compared 
with  the  power  which  the  machine  can  generate.  Suppose, 
therefore,  that  the  high-frequency  voltage  of  Fig.  140  were 
rectified.  It  would  then  give  a  voltage  and  current,  pulsating 


282  ELECTRICAL  APPARATUS 

with  the  frequency  of  the  exciting  current,  but  of  a  power,  as 
many  times  greater,  as  the  machine  output  is  greater  than  the 
exciting  power. 

Thus  such   an  inductor  alternator  with  alternating-current 
excitation  can  be  used  as  amplifier.     This    obviously    applies 
•equally  much  to  the  other  types,  as  shown  in  Figs.  136,  137 
and  138. 

Suppose  now  the  exciting  current  is  a  telephone  or  micro- 
phone current,  the  rectified  generated  current  then  pulsates  with 
the  frequencies  of  the  telephone  current,  and  the  machine  is  a 
telephonic  amplifier. 

Thus,  by  exciting  the  high-frequency  alternator  in  Fig.  138, 
by  a  telephone  current,  we  get  a  high-frequency  current,  of  an 
amplitude,  pulsating  with  the  telephone  current,  but  of  many 
times  greater  power  than  the  original  telephone  current.  This 
high-frequency  current,  being  of  the  frequency  suitable  for  radio 
communication,  now  is  sent  into  the  wireless  sending  antennae, 
and  the  current  received  from  the  wireless  receiving  antennae, 
rectified,  gives  wireless  telephonic  communications.  As  seen, 
the  power,  which  hereby  is  sent  out  from  the  wireless  antennae, 
is  not  the  insignificant  power  of  the  telephone  current,  but  is  the 
high-frequency  power  generated  by  the  alternator  with  telephonic 
excitation,  and  may  be  many  kilowatts,  thus  permitting  long- 
distance radio  telephony. 

It  is  obvious,  that  the  high  inductance  of  the  field  coil,  F,  of 
the  machine,  Fig.  138,  would  make  it  impossible  to  force  a  tele- 
phone current  through  it,  but  the  telephonic  exciting  current 
would  be  sent  through  the  armature  winding,  which  is  of  very 
low  inductance,  and  by  the  use  of  the  capacity  the  armature 
made  self-exciting  by  leading  current. 

Instead  of  sending  the  high-frequency  machine  current,  which 
pulsates  in  amplitude  with  telephonic  frequency,  through  radio 
transmission  and  rectifying  the  receiving  current,  we  can  rectify 
directly  the  generated  machine  current  and  so  get  a  current 
pulsating  with  the  telephonic  frequency,  that  is,  get  a  greatly 
amplified  telephone  current,  and  send  this  into  telephone  circuits 
for  long-distance  telephony. 

162.  Suppose,  now,  in  the  inductor  alternator,  Fig.  139,  with 
low-frequency  alternating-current  excitation,  giving  a  voltage 
wave  shown  in  Fig.  140,  we  use  several  alternators  excited  by 
low-frequency  currents  of  different  phases,  or  instead  of  a  single- 


INDUCTOR  MACHINES 


283 


phase  field,  as  in  Fig.  139,  we  use  a  polyphase  exciting  field.  This 
is  shown,  with  three  exciting  coils  or  poles  energized  by  three- 
phase  currents,  in  Fig.  141.  The  high-frequency  voltages  of 
pulsating  amplitude,  induced  by  the  three  phases,  then  super- 
pose a  high-frequency  wave  of  constant  amplitude,  and  we  get, 
in  Fig.  141,  a  high-frequency  alternator  with  polyphase  field 
excitation. 

Instead  of  using  definite  polar  projection  for  the  three-phase 
bipolar  exciting  winding,  as  shown  in  Fig.  141,  we  could  use  a 
distributed  winding,  like  that  in  an  induction  motor,  placed  in 
the  same  slots  as  the  inductor-alternator  armature  winding.  By 


FIG.  141. — Inductor  alternator  with  three-phase  excitation. 

placing  a  bipolar  short-circuited  winding  on  the  inductor,  the 
three-phase  exciting  winding  of  the  high-frequency  (24-polar) 
inductor  alternator  also  becomes  a  bipolar  induction-motor 
primary  winding,  supplying  the  power  driving  the  machine. 
That  is,  the  machine  is  a  combination  of  a  bipolar  induction 
motor  and  a  24-polar  inductor  alternator,  or  a  frequency 
converter. 

Instead  of  having  a  separate  high-frequency  inductor-alter- 
nator armature  winding,  and  low-frequency  induction  motor 
winding,  we  can  use  the  ^same  winding  for  both  purposes,  as 
shown  diagrammatically  in  Figs.  142  and  143.  The  stator 
winding,  Fig.  142,  bipolar,  or  four-polar  60-cycle,  is  a  low- 
frequency  winding,  for  instance,  has  one  slot  per  inductor  pole, 
that  is,  twice  as  many  slots  as  the  inductor  has  teeth.  Successive 
turns  then  differ  from  each  other  by  180°  in  phase,  for  the  high- 
frequency  inductor  voltage.  Thus  grouping  the  winding  in 


284 


ELECTRICAL  APPARATUS 


two  sections,  1  and  3,  and  2  and  4,  the  high-frequency  voltages 
in  the  two  sections  are  opposite  in  phase  from  each  other.  Con- 
necting, then,  as  shown  in  Fig.  143,  1  and  2  in  series,  and  4  and 
3  in  series  into  the  two  phases  of  the  quarter-phase  supply  cir- 
cuit, no  high-frequency  induction  exists  in  either  phase,  but  the 
high-frequency  voltage  is  generated  between  the  middle  points 


FIG.  142. — Induction  type  of  high-frequency  inductor  alternator. 

of  the  two  phases,  as  shown  in  Fig.  143,  and  we  thus  get  another 
form  of  a  frequency  converter,  changing  from  low-frequency 
polyphase  to  high-frequency  single-phase. 


FIG.  143. — Diagram  of  connection  of  induction  type  of  inductor  alternator. 

163.  A  type  of  inductor  machine,  very  extensively  used  in 
small  machines — as  ignition  dynamos  for  gasoline  engines — is 
shown  in  Fig.  144.  The  field,  F,  and  the  shuttle-shaped  armature, 
A,  are  stationary,  and  an  inductor,  /,  revolves  between  field  and 
armature,  and  so  alternately  sends  the  magnetic  field  flux  through 
the  armature,  first  in  one,  then  in  the  opposite  direction.  As 
seen,  in  this  type,  the  magnetic  flux  in  the  armature  reverses, 
by  what  may  be  called  magnetic  commutation.  Usually  in  these 


INDUCTOR  MACHINES 


285 


small  machines  the  field  excitation  is  not  by  direct  current,  but 
by  permanent  magnets. 

This  principle  of  magnetic  commutation,  that  is,  of  reversing 


FIG.  144. — Magneto  inductor  machine. 

the  magnetic  flux  produced  by  a  stationary  coil,  in  another 
stationary  coil  by  means  of  a  moving  "magneto  commutator" 
or  inductor,  has  been  extensively  used  in  single-phase  feeder 


FIG.  145. — Magneto  commutation  voltage  regulator. 

regulators,  the  so-called  "  magneto  regulators."  It  is  illustrated 
in  Fig.  145.  P  is  the  primary  coil  (shunt  coil  connected  across 
the  alternating  supply  circuit),  S  the  secondary  coil  (connected 
in  series  into  the  circuit  which  is  to  be  regulated) :  the  magnetic 
inductor,  7,  in  the  position  shown  in  drawn  lines  sends  the  mag- 


286 


ELECTRICAL  APPARATUS 


netic  flux  produced  by  the  primary  coil,  through  the  secondary 
coil,  in  the  direction  opposite  to  the  direction,  in  which  it  would 
send  the  magnetic  flux  through  the  secondary  coil  when  in  the 
position  I',  shown  in  dotted  lines.  In  vertical  position,  the 
inductor,  7,  would  pass  the  magnetic  flux  through  the  primary 
coil,  without  passing  it  through  the  secondary  coil,  that  is,  with- 
out inducing  voltage  in  the  secondary.  Thus  by  moving  the 
shuttle  or  inductor,  7,  from  position  7  over  the  vertical  position 
to  the  position  I',  the  voltage  induced  in  the  secondary  coil,  S, 
is  varied  from  maximum  boosting  over  to  zero  to  maximum 
lowering. 

164.  Fig.  146  shows  a  type  of  machine,  which  has  been  and 
still  is  used  to  some  extent,  for  alternators  as  well  as  for  direct- 


FIG.  146. — Semi-inductor  type  of  machine. 

current  commutating  machines,  and  which  may  be  called  an 
inductor  machine,  or  at  least  has  considerable  similarity  with  the 
inductor  type.  It  is  shown  in  Fig.  146  as  six-polar  machine, 
with  internal  field  and  external  armature,  but  can  easily  be  built 
with  internal  armature  and  external  field.  The  field  contains 
one  field  coil,,?7,  concentric  to  the  shaft.  The  poles  overhang  the 
field  coils,  and  all  poles  of  one  polarity,  N,  come  from  the  one 
side,  all  poles  of  the  other  polarity  from  the  other  side  of  the  field 
coil.  The  magnetic  structure  thus  consists  of  two  parts  which 
interlock  axially,  as  seen  in  Fig.  146. 

The  disadvantage  of  this  type  of  field  construction  is  the  high 
flux  leakage  between  the  field  poles,  which  tends  to  impair  the 
regulation  in  alternators,  and  makes  commutation  more  difficult 
for  direct-current  machines.  It  offers,  however,  the  advantage 


INDUCTOR  MACHINES  287 

of  simplicity  and  material  economy  in  machines  of  small  and 
moderate  size,  of  many  poles,  as  for  instance  in  small  very  low- 
speed  synchronous  motors,  etc. 

165.  In  its  structural  appearance,  inductor  machines  often 
have  a  considerable  similarity  with  reaction  machines.  The 
characteristic  difference  between  the  two  types,  however,  is, 
that  in  the  reaction  machine  voltage  is  induced  by  the  pulsation 
of  the  magnetic  flux  by  pulsating  reluctance  of  the  magnetic 
circuit  of  the  machine.  The  magnetic  pulsation  in  the  reaction 
machine  thus  extends  throughout  the  entire  magnetic  circuit 
of  the  machine,  and  if  direct-current  excitation  were  used,  the 
voltage  would  be  induced  in  the  exciting  circuit  also.  In  the 
inductor  machine,  however,  the  total  magnetic  flux  does  not 
pulsate,  but  is  constant,  and  no  voltage  is  induced  in  the  direct- 
current  exciting  circuit.  Induction  is  produced  in  the  armature 
by  shifting  the — constant — magnetic  flux  locally  from  armature 
coil  to  armature  coil.  The  important  problem  of  inductor 
alternator  design — and  in  general  of  the  design  of  magneto  com- 
mutation apparatus — is  to  have  the  shifting  of  the  magnetic 
flux  from  path  to  path  so  that  the  total  reluctance  and  thus  the 
total  magnetic  flux  does  not  vary,  otherwise  excessive  eddy- 
current  losses  would  result  in  the  magnetic  structure. 

It  is  interesting  to  note,  that  the  number  of  inductor  teeth  is 
one-half  the  number  of  poles.  An  inductor  with  p  projections 
thus  gives  twice  as  many  cycles  per  revolution,  thus  as  syn- 
chronous motor  would  run  at  half  the  speed  of  a  standard  syn- 
chronous machine  of  p  poles. 

As  the  result  hereof,  in  starting  polyphase  synchronpus 
machines  by  impressing  polyphase  voltage  on  the  armature  and 
using  the  hysteresis  and  the  induced  currents  in  the  field  poles, 
for  producing  the  torque  of  starting  and  acceleration,  there 
frequently  appears  at  half  synchronism  a  tendency  to  drop  into 
step  with  the  field  structure  as  inductor.  This  results  in  an 
increased  torque  when  approaching,  and  a  reduced  torque  when 
passing  beyond  half  synchronism,  thus  produces  a  drop  in  the 
torque  curve  and  is  iiable  to  produce  difficulty  in  passing  beyond 
half  speed  in  starting.  In  extreme  cases,  it  may  result  even  in 
a  negative  torque  when  passing  half  synchronism,  and  make  the 
machine  non-self-starting,  or  at  least  require  a  considerable 
increase  of  voltage  to  get  beyond  half  synchronism,  over  that 
required  to  start  from  rest. 


CHAPTER  XVIII 
SURGING  OF  SYNCHRONOUS  MOTORS 

166.  In  the  theory  of  the  synchronous  motor  the  assumption 
is  made  that  the  mechanical  output  of  the  motor  equals  the  power 
developed  by  it.  This  is  the  case  only  if  the  motor  runs  at 
constant  speed.  If,  however,  it  accelerates,  the  power  input  is 
greater;  if  it  decelerates,  less  than  the  power  output,  by  the  power 
stored  in  and  returned  by  the  momentum.  Obviously,  the 
motor  can  neither  constantly  accelerate  nor  decelerate,  without 
breaking  out  of  synchronism. 

If,  for  instance,  at  a  certain  moment  the  power  produced  by 
the  motor  exceeds  the  mechanical  load  (as  in  the  moment  of 
throwing  off  a  part  of  the  load),  the  excess  power  is  consumed  by 
the  momentum  as  acceleration,  causing  an  increase  of  speed. 
The  result  thereof  is  that  the  phase  of  the  counter  e.m.f.,  e, 
is  not  constant,  but  its  vector,  e,  moves  backward  to  earlier  time, 
or  counter-clockwise,  at  a  rate  depending  upon  the  momentum. 
Thereby  the  current  changes  and  the  power  developed  changes 
and  decreases.  As  soon  as  the  power  produced  equals  the  load, 
the  acceleration  ceases,  but  the  vector,  e,  still  being  in  motion, 
due  to  the  increased  speed,  further  reduces  the  power,  causing 
a  retardation  and  thereby  a  decrease  of  speed,  at  a  rate  depend- 
ing upon  the  mechanical  momentum.  In  this  manner  a  periodic 
variation  of  the  phase  relation  between  e  and  e0,  and  correspond- 
ing variation  of  speed  and  current  occurs,  of  an  amplitude  and 
period  depending  upon  the  circuit  conditions  and  the  mechanical 
momentum. 

If  the  amplitude  of  this  pulsation  has  a  positive  decrement, 
that  is,  is  decreasing,  the  motor  assumes  after  a  while  a  constant 
position  of  e  regarding  e0,  that  is,  its  speed  becomes  uniform. 
If,  however,  the  decrement  of  the  pulsation  is  negative,  an 
infinitely  small  pulsation  will  continuously  increase  in  amplitude, 
until  the  motor  is  thrown  out  of  step,  or  the  decrement  becomes 
zero,  by  the  power  consumed  by  forces  opposing  the  pulsation, 
as  anti-surging  devices,  or  by  the  periodic  pulsation  of  the  syn- 
chronous reactance,  etc.  If  the  decrement  is  zero,  a  pulsation 

288 


SURGING  OF  SYNCHRONOUS  MOTORS          289 

started  once  will  continue  indefinitely  at  constant  amplitude. 
This  phenomenon,  a  surging  by  what  may  be  called  electro- 
mechanical resonance,  must  be  taken  into  consideration  in  a 
complete  theory  of  the  synchronous  motor. 
167.  Let: 

EQ  =  e0  =  impressed  e.m.f.  assumed  as  zero  vector. 
E    =  e  (cos  |8  —  j  sin  ft  =  e.m.f.  consumed  by  counter  e.m.f. 
of  motor,  where : 

|8  =  phase  angle  between  E0  and  E. 
Let: 

Z  =  r  +  jx, 
and  z  =  \/r2  +  x2 

=  impedance  of  circuit  between 
Eo  and  E,  and 

tan  a  =  — 

r  ; 

The  current  in  the  system  is: 

_  Co  —  E  _  e0  —  e  cos  |8  +  je  sin  ff 
Tt*  ~TT7*~ 

=  -  {[e<>  cos  a  —  e  cos  (a  +  ft] 

—  j  [e0  sin  a  —  e  sin  (a  +  0)]}        (1) 
The  power  developed  by  the  synchronous  motor  is : 

Po  =  [El]1  =  -  {[cos  0  [e0  cos  a  -  e  cos  (a  +  ft] 

+  sin  0  [e0  sin  a  —  e  sin  (a  +  ft] } 

=  -  { [e0  cos  (a  —  ft  —  e  cos  a] } .       (2) 

If,  now,  a  pulsation  of  the  synchronous  motor  occurs,  resulting 
in  a  change  of  the  phase  relation,  0,  between  the  counter  e.m.f.,  e, 
and  the  impressed  e.m.f.,  eQ  (the  latter  being  of  constant  fre- 
quency, thus  constant  phase),  by  an  angle,  6,  where  8  is  a  periodic 
function  of  time,  of  a  frequency  very  low  compared  with  the 
impressed  frequency,  then  the  phase  angle  of  the  counter  e.m.f., 
e,  is  0  +  6;  and  the  counter  e.m.f.  is: 

\E  =  e  {cos  (0  +  5)  -  j  sin  (0  +  5)j, 

19 


290  ELECTRICAL  APPARATUS 

hence  the  current: 

/  =  -  ([0o  cos  a  —  e  cos  (a  +  |8  +  5)] 
—  j  [e0  sin  a  —  e  sin  (a  +  0  +  6)] } 
=  7o  +  ™  sin  ^  {sin (a  +  /5  +  |)  +  j  cos(a  +  0  +  ~)  }  (3) 


the  power: 


/> 

P  =  -  je0  cos  (a  —  j8  —  6)  —  e  cos  a 


Let  now: 

VQ  =  mean  velocity   (linear,   at  radius  of  gyration)   of    syn- 

chronous machine; 
s    =  slip,  or  decrease  of  velocity,  as  fraction  of  v0,  where  s  is 

a  (periodic)  function  of  time;  hence 
v    =  VQ  (1  —  s)  '  —  actual  velocity,  at  time,  t. 

During  the  time  element,  dt,  the  position  of  the  synchronous 
motor  armature  regarding  the  impressed  e.m.f.,  e0,  and  thereby 
the  phase  angle,  0  +  5,  of  e,  changes  by: 

dd  =  2  irfsdt 

=  sdB,  (5) 

where  : 

0  =  2  irft, 
and 

/  =  frequency  of  impressed  e.m.f.,  eQ. 
Let: 

m     =  mass  of  revolving  machine  elements,  and 

MQ  =  %  mv<?   =   mean  mechanical  momentum,  reduced  to 

joules   or  watt-seconds;  then  the  momentum  at  time,  t,  and 

velocity  v  =  v0  (I  —  s)  is: 


M  =  K  ^o2  (1  -  s)2, 

and  the  change  of  momentum  during  the  time  element,  dt,  is: 
dM  _   .  ds 


SURGING  OF  SYNCHRONOUS  MOTORS  291 

hence,  for  small  values  of  s: 

dM  ,ds  dd 

=~- 


n  ,,  ds  dd 
~2M°« 
Since: 


and  from  (5)  : 

dd 

s  =  de' 

ds  =  d25 
dS  ~  dO2 
it  is: 

dM 


dt  ~"J^"dB'  ^ 

Since,  as  discussed,  the  change  of  momentum  equals  the  dif- 
ference between  produced  and  consumed  power,  the  excess  of 
power  being  converted  into  momentum,  it  is : 

P  -  PQ  =  dM,  (8) 

and,  substituting  (4)  and  (7)  into  (8)  and  rearranging : 

-^  sin  I  sin  (a  -  ft  -  |)  +  2  rfM.  g  =  0.  (9) 

Assuming  5  as  a  small  angle,  that  is,  considering  only  small 
oscillations,  it  is : 

.     5^5 
n  2  ~  2' 

'  sin  \a  —  |3  —  »    =  sin  (a  —  0) ; 
hence,  substituted  in  (18) : 

~-  6  sin  (a  —  |8)  +  4  vfM o  ^  =  0,  (10) 

and,  substituting: 

eco  sin  (a  —  3)  /^  ^  \ 

—  (11) 


it  is: 

.    d*d 


292  ELECTRICAL  APPARATUS 

This  differential  equation  is  integrated  by  : 

5  =  Aeco,  (13) 

which,  substituted  in  (12)  gives: 

aAec0  +  AC2€ce  =  0, 
a  +  C2  =  0, 

C  =  ±  V-a. 


168.  1.  If  a  <  0,  it  is: 
where  : 


s  A       +m8 

5  =  Aie 


/  -  /      ee0  sin  (ft  -  a) 

m  =  v  —  a 


Since  in  this  case,  e+m9  is  continually  increasing,  the  syn- 
chronous motor  is  unstable.  That  is,  without  oscillation,  the 
synchronous  motor  drops  out  of  step,  if  ft  >  a. 

2.  If  a  >  0,  it  is,  denoting: 


in  (a  - 
n 


6    -    A 

or,  substitutmg  for  c+ine  and  e+JH°  the  trigonometric  functions: 

6  =  (Ai  +  A2)  cos  nd  +  j  (Ai  —  A2)  sin  n0, 
or, 

6  =  B  cos  (nd  +  T). 


That  is,  the  synchronous  motor  is  in  stable  equilibrium,  when 
oscillating  with  a  constant  amplitude  B,  depending  upon  the 
initial  conditions  of  oscillation,  and  a  period,  which  for  small 
oscillations  gives  the  frequency  of  oscillation  : 


,  ,          / 

=  nf  '-  V 


/eeosin  (a  —  ft) 


np. 


As  instance,  let: 

e0  =  2200  volts.     Z  =   1  +  4  j  ohms,  or,  z  =  4.12;  a  =  76°. 

And  let  the  machine,  a  16-polar,  60-cycle,  400-kw.,  revolving- 
field,  synchronous  motor,  have  the  radius  of  gyration  of  20  in., 
a  weight  of  the  revolving  part  of  6000  Ib. 

The  momentum  then  is  M0  =  850,000  joules. 

Deriving  the  angles,  ft,  corresponding  to  given  values  of  output, 
P,  and  excitation,  e,  from  the  polar  diagram,  or  from  the  symbolic 


SURGING  OF  SYNCHRONOUS  MOTORS          293 

representation,  and  substituting  in  (16),  gives  the  frequency  of 
oscillation : 

P  =  0: 

e   =  1600  volts;  0  =  -  2°;/0  =  2.17  cycles, 

or  130  periods  per  minute. 
2180Tolts  +3°  2.50  cycles, 

or  150  periods  per  minute. 
2800  volts  +  5°  2.85  cycles, 

or  169  periods  per  minute. 
P  =  400  kw. 
e   =  1600  volts;  0  =  33°; /„  =  1.90  cycles, 

or  114  periods  per  minute. 
2180  volts  21°  2.31  cycles, 

or  139  periods  per  minute. 
2800  volts  22°  2.61  cycles, 

or  154  periods  per  minute. 

As  seen,  the  frequency  of  oscillation  does  not  vary  much  with 
the  load  and  with  the  excitation.  It  slightly  decreases  with 
increase  of  load,  and  it  increases  with  increase  of  excitation. 

In  this  instance,  only  the  momentum  of  the  motor  has  been 
considered,  as  would  be  the  case  for  instance  in  a  synchronous 
converter. 

In  a  direct-connected  motor-generator  set,  assuming  the 
momentum  of  the  direct-current-generator  armature  equal  to 
60  per  cent,  of  the  momentum  of  the  synchronous  motor,  the 
total  momentum  is  M0  =  1,360,000  joules,  hence,  at  no-load: 

P  =  0, 

e  =  1600  volts  ;/0  =  1.72  cycles,  or  103  periods  per  minute. 
1.98  cycles,  or  119  periods  per  minute. 
1.23  cycles,  or  134  periods  per  minute. 

.  169.  In  the  preceding  discussion  of  the  surging  of  synchronous 
machines,  the  assumption  has  been  made  that  the  mechanical 
power  consumed  by  the  load  is  constant,  and  that  no  damping 
or  anti-surging  deVices  were  used. 

The  mechanical  power  consumed  by  the  load  varies,  however, 
more  or  less  with  the  speed,  approximately  proportional  to  the 
speed  if  the  motor  directly  drives  mechanical  apparatus,  as 
pumps,  etc.,  and  at  a  higher  power  of  the  speed  if  driving  direct- 
current  generators,  or  as  synchronous  converter,  especially 


294  ELECTRICAL  APPARATUS 

when  in  parallel  with  other  direct-current  generators.  Assum- 
ing, then,  in  the  general  case  the  mechanical  power  consumed  by 
the  load  to  vary,  within  the  narrow  range  of  speed  variation  con- 
sidered during  the  oscillation,  at  the  pth  power  of  the  speed, 
in  the  preceding  equation  instead  of  P0  is  to  be  substituted, 
Po(l  -s)p  =  Po(l  -  PS). 

If  anti-surging  devices  are  used,  and  even  without  these  in 
machines  in  which  eddy  currents  can  be  produced  by  the  oscilla- 
tion of  slip,  in  solid  field  poles,  etc.,  a  torque  is  produced  more 
or  less  proportional  to  the  deviation  of  speed  from  synchronism. 
This  power  assumes  the  form,  PI  =  c2s,  where  c  is  a  function  of 
the  conductivity  of  the  eddy-current  circuit  and  the  intensity 
of  the  magnetic  field  of  the  machine,  c2  is  the  power  which 
would  be  required  to  drive  the  magnetic  field  of  the  motor 
through  the  circuits  of  the  anti-surging  device  at  full  frequency, 
if  the  same  relative  proportions  could  be  retained  at  full  fre- 
quency as  at  the  frequency  of  slip,  s.  That  is,  PI  is  the  power 
produced  by  the  motor  as  induction  machine  at  slip  s.  In- 
stead of  P,  the  power  generated  by  the  motor,  in  the  preced- 
ing equations  the  value,  P  +  PI,  has  to  be  substituted,  then: 

The  equation  (8)  assumes  the  form: 


or: 

(P  -  P.)  -  (P!  +  pPos)  =  dft<  (17) 

or,  substituting  (7)  and  (4)  : 

2«  Jrin  J  Bin  [„-/}-  J]+  (C«  +  pP0)  g  +  4*fM0  g  =  0; 

(18) 
and,  for  small  values  of  d: 


b  =  .  (20) 

8  TTjM  o 

Of  these  two  terms  b  represents  the  consumption,  a  the  oscilla- 
tion of  energy  by  the  pulsation  of  phase  angle,  0.     b  and  a  thus 


SURGING  OF  SYNCHRONOUS  MOTORS  295 

have  a  similar  relation  as  resistance  and  reactance  in  alternating- 
current  circuits,  or  in  the  discharge  of  condensers,  a  is  the 
same  term  as  in  paragraph  167. 

Differential  equation  (19)  is  integrated  by: 

8  =  Atce,  (21) 

which,  substituted  in  (19),  gives: 

aAece  +•  2  bCAece  +  C2Aece  =  0, 
a  +  2  bC  +  C2  =  0, 
which  equation  has  the  two  roots: 


Ci  =  -b  +  \/b2  -  a, 

C2  =  -b  -  -V/&2  -  a-  (22) 

1.  If  a  <  0,  or  negative,  that  is  0  >  a,  Ci  is  positive  and  C2 
negative,  and  the  term  with  C\  is  continuously  increasing,  that 
is,  the  synchronous  motor  is  unstable,  and,  without  oscillation, 
drifts  out  of  step. 

2.  If  0  <  a  <  b2,  or  a  positive,  and  b2  larger  than  a  (that  is, 
the  energy-consuming  term  very  large),   C\  and  C2  are  both 
negative,  and,  by  substituting,  +  \/62  —  o  =  flf,  it  is: 

Ci=-(b-3),        Ci=-(b  +  Jf); 
hence  : 

6  =  A1e-(6-")^  +  ^2€-(6  +  ^-  (23) 

That  is,  the  motor  steadies  down  to  its  mean  position  logarith- 
mically, or  without  any  oscillation. 

62  >  a, 
hence: 

(c2  +  pP0)2  ^     eeQ  sin  (a  -  0) 

2 


is  the  condition  under  which  no  oscillation  can  occur. 

As  seen,  the  left  side  of  (24)  contains  only  mechanical,  the 
right  side  only  electrical  terms. 

3.  a  >  b\ 

In  this  case,  \/62  —  a  is  imaginary,  and,  substituting: 


g    =       a  -  b2, 
it  is: 

Ci  =  -b+jg, 
C2  =  -b-jg, 


296  ELECTRICAL  APPARATUS 

hence  : 


and,  substituting  the  trigonometric  for  the  exponential  functions, 
gives  ultimately: 

5  =  £e-6'cos  (00  +  7).  (25) 

That  is,  the  motor  steadies  down  with  an  oscillation  of  period  : 


=       fee,  sin  (a  -  ft)  _  (c2  +  pP0)2 
"  \         4  TrzMo  64 

and  decrement  or  attenuation  constant: 


b  =  .  (27) 

8 


170.  It  follows,  however,  that  under  the  conditions  considered, 
a  cumulative  surging,  or  an  oscillation  with  continuously  increas- 
ing amplitude,  can  not  occur,  but  that  a  synchronous  motor, 
when  displaced  in  phase  from  its  mean  position,  returns  thereto 
either  aperiodically,  if  62  >  a,  or  with  an  oscillation  of  vanishing 
amplitude,  if  62  <  a.  At  the  worst,  it  may  oscillate  with  constant 
amplitude,  if  b  =  0. 

t  Cumulative  surging  can,  therefore,  occur  only  if  in  the  differ- 
ential equation  (19)  : 

"  +  "s  +  S-°'  '  (28) 

the  coefficient,  6,  is  negative. 

Since  c2,  representing  the  induction  motor  torque  of  the  damp- 
ing device,  etc.,  is  positive,  and  pPo  is  also  positive  (p  being 
the  exponent  of  power  variation  with  speed),  this  presupposes 

—  h2 
the  existence  of  a  third  and  negative  term,  5  —  7^,  in  6: 


•  (29) 

This  negative  term  represents  a  power: 

P2  =  -h*s;  (30) 

that  is,  a  retarding  torque  during  slow  speed,  or  increasing  |8,  and 
accelerating  torque  during  high  speed,  or  decreasing  0. 

The  source  of  this  torque  may  be  found  external  to  the  motor, 
or  internal,  in  its  magnetic  circuit. 


SURGING  OF  SYNCHRONOUS  MOTORS          297 

External  sources  of  negative,  P2,  may  be,  for  instance,  the 
magnetic  field  of  a  self-exciting,  direct-current  generator,  driven 
by  the  synchronous  motor.  With  decrease  of  speed,  this  field 
decreases,  due  to  the  decrease  of  generated  voltage,  and  increases 
with  increase  of  speed.  This  change  of  field  strength,  however, 
lags  behind  the  exciting  voltage  and  thus  speed,  that  is,  during 
decrease  of  speed  the  output  is  greater  than  during  increase  of 
speed.  If  this  direct-current  generator  is  the  exciter  of  the 
synchronous  motor,  the  effect  may  be  intensified. 

The  change  of  power  input  into  the  synchronous  motor,  with 
change  of  speed,  may  cause  the  governor  to  act  on  the  prime 
mover  driving  the  generator,  which  supplies  power  to  the  motor, 
and  the  lag  of  the  governor  behind  the  change  of  output  gives  a 
pulsation  of  the  generator  frequency,  of  e0,  which  acts  like 
a  negative  power,  P2.  The  pulsation  of  impressed  voltage, 
caused  by  the  pulsation  of  |8,  may  give  rise  to  a  negative, 
P2,  also. 

An  internal  cause  of  a  negative  term,  P2,  is  found  in  the  lag 
of  the  synchronous  motor  field  behind  the  resultant  m.m.f.  In 
the  preceding  discussion,  e  is  the  "  nominal  generated  e.m.f." 
of  the  synchronous  machine,  corresponding  to  the  field  excita- 
tion. The  actual  magnetic  flux  of  the  machine,  however,  does 
not  correspond  to  e,  and  thus  to  the  field  excitation,  but  corre- 
sponds to  the  resultant  m.m.f.  of  field  excitation  and  armature 
reaction,  which  latter  varies  in  intensity  and  in  phase  during  the 
oscillation  of  0.  Hence,  while  e  is  constant,  the  magnetic  flux 
is  not  constant,  but  pulsates  with  the  oscillations  of  the  machine. 
This  pulsation  of  the  magnetic  flux  lags  behind  the  pulsation  of 
m.m.f.,  and  thereby  gives  rise  to  a  term  in  b  in  equation  (28). 
If  Po,  ]8,  e,  60,  Z  are  such  that  a  retardation  of  the  motor  increases 
the  magnetizing,  or  decreases  the  demagnetizing  force  of  the 
armature  reaction,  a  negative  term,  P2,  appears,  otherwise  a 
positive  term. 

P2  in  this  case  is  the  energy  consumed  by  the  magnetic  cycle 
of  the  machine  at  full  frequency,  assuming  the  cycle  at  full  fre- 
quency as  the  same  as  at  frequency  of  slip,  s. 

Or  inversely,  e  may  be  said  to  pulsate,  due  to  the  pulsation  of 
armature  reaction,  with  the  same  frequency  as  j8,  but  with  a 
phase,  which  may  either  be  lagging  or  leading.  Lagging  of  the 
pulsation  of  e  causes  a  negative,  leading  a  positive,  P2. 

P2,  therefore,  represents  the  power  due  to  the  pulsation  of  e 


298  ELECTRICAL  APPARATUS 

caused  by  the  pulsation  of  the  armature  reaction,  as  discussed  in 
"  Theory  and  Calculation  of  Alternating-Current  Phenomena." 

Any  appliance  increasing  the  area  of  the  magnetic  cycle  of 
pulsation,  as  short-circuits  around  the  field  poles,  therefore, 
increases  the  steadiness  of  a  steady  and  increases  the  unsteadi- 
ness of  an  unsteady  synchronous  motor. 

"In  self  -exciting  synchronous  converters,  the  pulsation  of  e  is 
intensified  by  the  pulsation  of  direct-current  voltage  caused 
thereby,  and  hence  of  excitation. 

Introducing  now  the  term,  P2  =  —  h2s,  into  the  differential 
equations  of  paragraph  169,  gives  the  additional  cases: 

b  <  0,  or.  negative,  that  is  : 

c^fpPo-J» 

87r/Mo 

Hence,  denoting: 

,     (32) 


8 

gives  : 

4.  If:  6i2  >  a,  g  =  +  VV  -  a, 

5  =  Aie  +  (6'+/)'  +  A,6  +  (6'-/)a.  (33) 

That  is,  without  oscillation,  the  motor  drifts  out  of  step,  in 
unstable  equilibrium. 

5.  If:  a  >  Zh2,  g  =  vV^&i2, 


s(g8  +  5).  (34) 

That  is,  the  motor  oscillates,  with  constantly  increasing  am- 
plitude, until  it  drops  out  of  step.  This  is  the  typical  case  of 
cumulative  surging  by  electro-mechanical  resonance. 

The  problem  of  surging  of  synchronous  machines,  and  its 
elimination,  thus  resolves  into  the  investigation  of  the  coefficient  : 

.   _.c«  +  pP.-ft' 

Sx/Mo 

while  the  frequency  of  surging,  where  such  exists,  is  given  by: 

,   .        I  fee,  sin  (a  -  ft)          (c2  +  pP0  -~h*)~*  ,_, 

V          47TZoM0   "  647rWo2 


Case  (4),  steady  drifting  out  of  step,  has  only  rarely  been 
observed. 

The  avoidance  of  surging  thus  requires: 


SURGING  OF  SYNCHRONOUS  MOTORS  299 

1.  An  elimination  of  the  term  h2,  or  reduction  as  far  as  possible. 

2.  A  sufficiently  large  term,  c2,  or 

3.  A  sufficiently  large  term,  pP0. 

(I)  refers  to  the  design  of  the  synchronous  machine  and  the 
system  on  which  it  operates.  (2)  leads  to  the  use  of  electro- 
magnetic anti-surging  devices,  as  an  induction  motor  winding  in 
the  field  poles,  short-circuits  between  the  poles,  or  around  the 
poles,  and  (3)  leads  to  flexible  connection  to  a  load  or  a  mo- 
mentum, as  flexible  connection  with  a  flywheel,  or  belt  drive  of 
the  load. 

The  conditions  of  steadiness  are: 

0  >  «, 

c2  +  pP0  ~  h2  >  0, 
and  if: 

(c2  +  pPo  -  ft2)2  .     ee0  sin  (a  -  ft) 

16  TT/Mo  Z 

no  oscillation  at  all  occurs,  otherwise  an  oscillation  with  decreas- 
ing amplitude. 

As  seen,  cumulative  oscillation,  that  is,  hunting  or  surging, 
can  occur  only,  if  there  is  a  source  of  power  supply  converting 
into  low-frequency  pulsating  power,  and  the  mechanism  of  con- 
version is  a  lag  of  some  effect — in  the  magnetic  field  of  the 
machine,  or  external — which  causes  the  forces  restoring  the 
machine  into  step,  to  be  greater  than  the  forces  which  oppose  the 
deviation  from  the  position  in  step  corresponding  to  the  load, 
For  further  discussion  of  the  phenomenon  of  cumulative  surging, 
and  of  cumulative  oscillations  in  general,  see  Chapter  XI  of 
"Theory  and  Calculation  of  Electric  Circuits." 


CHAPTER  XIX 
ALTERNATING-CURRENT  MOTORS  IN  GENERAL 

171.  The  starting  point  of  the  theory  of  the  polyphase  and 
single-phase  induction  motor  usually  is  the  general  alternating- 
current  transformer.  Coming,  however,  to  the  commutator 
motors,  this  method  becomes  less  suitable,  and  the  following 
more  general  method  preferable. 

In  its  general  form  the  alternating-current  motor  consists  of 
one  or  more  stationary  electric  circuits  magnetically  related  to 
one  or  more  rotating  electric  circuits.  These  circuits  can  be 
excited  by  alternating  currents,  or  some  by  alternating,  others 
by  direct  current,  or  closed  upon  themselves,  etc.,  and  connec- 
tion can  be  made  to  the  rotating  member  either  by  collector 
rings — that  is,  to  fixed  points  of  the  windings — or  by  commutator 
—that  is,  to  fixed  points  in  space. 

The  alternating-current  motors  can  be  subdivided  into  two 
classes — those  in  which  the  electric  and  magnetic  relations 
between  stationary  and  moving  members  do  not  vary  with  their 
relative  positions,  and  those  in  which  they  vary  with  the  relative 
positions  of  stator  and  rotor.  In  the  latter  a  cycle  of  rotation 
exists,  and  therefrom  the  tendency  of  the  motor  results  to  lock  at 
a  speed  giving  a  definite  ratio  between  the  frequency  of  rotation 
and  the  frequency  of  impressed  e.m.f.  Such  motors,  therefore, 
are  synchronous  motors. 

The  main  types  of  synchronous  motors  are  as  follows : 

1.  One  member  supplied  with  alternating  and  the  other  with 
direct  current — polyphase  or  single-phase  synchronous  motors. 

2.  One  member  excited  by  alternating  current,  the  other  con- 
taining a  single  circuit  closed  upon  itself — synchronous  induction 
motors. 

3.  One  member  excited  by  alternating  current,  the  other  of 
different  magnetic  reluctance  in  different  directions   (as  polar 
construction) — reaction  motors. 

4.  One  member  excited  by  alternating  current,  the  other  by 
alternating  current  of  different  frequency  or  different  direction 
of    rotation — general    alternating-current    transformer    or    fre- 
quency converter  and  synchronous-induction  generator. 

300 


ALTERNATING-CURRENT  MOTORS  301 

(1)  is  the  synchronous  motor  of  the  electrical  industry.     (2) 
and  (3)  are  used  occasionally  to  produce  synchronous  rotation 
without  direct-current  excitation,  and  of  very  great  steadiness 
of  the  rate  of  rotation,   where   weight  efficiency  and  power- 
factor  are  of  secondary  importance.     (4)  is  used  to  some  extent 
as  frequency  converter  or  alternating-current  generator. 

(2)  and  (3)  are  occasionally  observed  in  induction  machines, 
and  in  the  starting  of  synchronous  motors,  as  a  tendency  to 
lock  at  some  intermediate,  occasionally  low,  speed.     That  is, 
in  starting,  the  motor  does  not  accelerate  up  to  full  speed,  but 
the  acceleration  stops  at  some  intermediate  speed,  frequently 
half  speed,  and  to  carry  the  motor  beyond  this  speed,  the  im- 
pressed voltage  may  have  to  be  raised  or  even  external  power 
applied.     The  appearance  of  such  "dead  points"  in  the  speed 
curve  is   due  to  a  mechanical  defect — as  eccentricity  of  the 
rotor — or  faulty  electrical  design:  an  improper  distribution  of 
primary  and  secondary  windings  causes  a  periodic  variation  of 
the  mutual  inductive  reactance  and  so  of  the  effective  primary 
inductive  reactance,  (2)  or  the  use  of  sharply  defined  and  im- 
properly  arranged   teeth  in   both   elements   causes   a   periodic 
magnetic  lock  (opening  and  closing  of.  the  magnetic  circuit,  (3) 
and  so  a  tendency  to  synchronize  at  the  speed  corresponding  to 
this  cycle. 

Synchronous  machines  have  been  discussed  elsewhere.  Here 
shall  be  considered  only  that  type  of  motor  in  which  the  electric 
and  magnetic  relations  between  the  stator  and  rotor  do  not  vary 
with  their  relative  positions,  and  the  torque  is,  therefore,  riot 
limited  to  a  definite  synchronous  speed.  This  requires  that  the 
rotor  when  connected  to  the  outside  circuit  be  connected  through 
a  commutator,  and  when  closed  upon  itself,  several  closed  cir- 
cuits exist,  displaced  in  position  from  each  other  so  as  to  offer  a 
resultant  closed  circuit  in  any  direction. 

The  main  types  of  these  motors  are: 

1.  One  member  supplied  with  polyphase  or  single-phase  alter- 
nating voltage,  the  other  containing  several  circuits  closed  upon 
themselves — polyphase  and  single-phase  induction  machines. 

2.  One  member  supplied  with  polyphase  or  single-phase  alter- 
nating voltage,  the  other  connected  by  a  commutator  to  an 
alternating  voltage — compensated  induction  motors,  commutator 
motors  with  shunt-motor  characteristic. 

3.  Both  members  connected,  through  a  commutator,  directly 


302  ELECTRICAL  APPARATUS 

or  inductively,  in  series  with  each  other,  to  an  alternating  vol- 
tage— alternating-current  motors  with  series-motor  characteristic. 

Herefrom  then  follow  three  main  classes  of  alternating-current 
motors : 

Synchronous  motors. 

Induction  motors. 

Commutator  motors. 

There  are,  however,  numerous  intermediate  forms,  which 
belong  in  several  classes,  as  the 'synchronous-induction  motor, 
the  compensated-induction  motor,  etc. 

172.  An  alternating  current,  7,  in  an  electric  circuit  produces 
a  magnetic  flux,  <£,  interlinked  with  this  circuit.  Considering 
equivalent  sine  waves  of  7  and  3>,  3>  lags  behind  I  by  the  angle 
of  hysteretic  lag,  a.  This  magnetic  flux,  3>,  generates  an  e.m.f., 
E  =  2  Tr/n^,  where  /  =  frequency,  n  =  number  of  turns  of 
electric  circuit.  This  generated  e.m.f.,  E,  lags  90°  behind  the 
magnetic  flux,  <£,  hence  consumes  an  e.m.f.  90°  ahead  of  <l>, 
or  90 — a  degrees  ahead  of  I.  This  may  be  resolved  in  a  reactive 
component:  E  =  2irfn$  cos  a  =  2  irfLI  =  xl,  the  e.m.f.  con- 
sumed by  self-induction,  and  power  component:  E"  =  2irfn^ 
sin  a  =  2irfHI  =  r"I  =, e.m.f.  consumed  by  hysteresis  (eddy 
currents,  etc.),  and  is,  therefore,  in  vector  representation  denoted 
by: 

W  =  jxj  and  $"  =  r"I, 
where : 

x  =  2  TT/L  =  reactance, 
and 

L  =  inductance, 

r"  =  effective  hysteretic  resistance. 

The  ohmic  resistance  of  the  circuit,  r',  consumes  an  e.m.f. 
r'l,  in  phase  with  the  current,  and  the  total  or  effective  resistance 
of  the  circuit  is,  therefore,  r  =  r'  +  r",  and  the  total  e.m.f. 
consumed  by  the  circuit,  or  the  impressed  e.m.f.,  is: 

E  =  (r+jx)f  =  Z!, 
where : 

Z  =  r  +  jx  =  impedance,  in  vector  denotation, 
z  =  vV2  +  x2  =  impedance,  in  absolute  terms. 

If  an  electric  circuit  is  in  inductive  relation  to  another  electric 
circuit,  it  is  advisable  to  separate  the  inductance,  L,  of  the  cir- 


ALTERNATING-CURRENT  MOTORS  303 

cuit  in  two  parts — the  self-inductance,  S,  which  refers  to  that 
part  of  the  magnetic  flux  produced  by  the  current  in  one  circuit 
which  is  interlinked  only  with  this  circuit  but  not  with  the  other 
circuit,  and  the  mutual  inductance,  M,  which  refers  to  that  part 
of  the  magnetic  flux  interlinked  also  with  the  second  circuit. 
The  desirability  of  this  separation  results  from  the  different  char- 
acter of  the  two  components:  The  self-inductive  reactance  gen- 
erates a  reactive  e.m.f.  and  thereby  causes  a  lag  of  the  current, 
while  the  mutual  inductive  reactance  transfers  power  into  the 
second  circuit,  hence  generally  does  the  useful  work  of  the  ap- 
paratus. This  leads  to  the  distinction  between  the  self-inductive 
impedance,  Z0  =  r0  +  jxQ,  and  the  mutual  inductive  impedance, 
Z  =  r  +  jx. 

The  same  separation  of  the  total  inductive  reactance  into  self- 
inductive  reactance  and  mutual  inductive  reactance,  represented 
respectively  by  the  self-inductive  or  "leakage"  impedance,  and 
the  mutual  inductive  or  "exciting"  impedance  has  been  made 
in  the  theory  of  the  transformer  and  the  induction  machine.  In 
those,  the  mutual  inductive  reactance  has  been  represented,  not 
by  the  mutual  inductive  impedance,  Z,  but  by  its  reciprocal 

value,  the  exciting  admittance :  Y  =  -^-     It  is  then : 

r0  is  the  coefficient  of  power  consumption  by  ohmic  resistance, 
hysteresis  and  eddy  currents  of  the  self-inductive  flux — effective 
resistance. 

xQ  is  the  coefficient  of  e.m.f.  consumed  by  the  self-inductive  or 
leakage  flux — self-inductive  reactance. 

r  is  the  coefficient  of  power  consumption  by  hysteresis  and 
eddy  currents  due  to  the  mutual  magnetic  flux  (hence  contains 
no  ohmic  resistance  component). 

x  is  the  coefficient  of  e.m.f.  consumed  by  the  mutual  magnetic 
flux. 

The  e.m.f.  consumed  by  the  circuit  is  then: 

E  =  Z»I  +  ZJ.       i  (1) 

If  one  of  the  circuits  rotates  relatively  to  the  other,  then  in 
addition  to  the  e.m.f.  of  self-inductive  impedance :  ZQI,  and  the 
e.m.f.  of  mutual-inductive  impedance  or  e.m.f.  of  alternation: 
ZI,  an  e.m.f.  is  consumed  by  rotation.  This  e.m.f.  is  in  phase 
with  the  flux  through  which  the  coil  rotates — that  is,  the  flux 
parallel  to  the  plane  of  the  coil — and  proportional  to  the  speed — 


304 


ELECTRICAL  APPARATUS 


that  is,  the  frequency  of  rotation — while  the  e.m.f.  of  alternation 
is  90°  ahead  of  the  flux  alternating  through  the  coil — that  is,  the 
flux  parallel  to  the  axis  of  the  coil — and  proportional  to  the  fre- 
quency. If,  therefore,  Z'  is  the  impedance  corresponding  to  the 
former  flux,  the  e.m.f.  of  rotation  is  —jSZ'I,  where  S  is  the 
ratio  of  frequency  of  rotation  to  frequency  of  alternation,  or  the 
speed  expressed  in  fractions  of  synchronous  speed.  The  total 
e.m.f.  consumed  in  the  circuit  is  thus : 

E  =  Z0I  +  ZI  -  JSZ'I.  (2) 

Applying  now  these  considerations  to  the  alternating-current 
motor,  we  assume  all  circuits  reduced  to  the  same  number  of 
turns — that  is,  selecting  one  circuit,  of  n  effective  turns,  as  start- 
ing point,  if  Hi  =  number  of  effective  turns  of  any  other  circuit, 
all  the  e.m.fs.  of  the  latter  circuit  are  divided,  the  currents  multi- 
plied with  the  ratio,  —  >  the  impedances  divided,  the  admittances 


This  reduction  of  the  constants  of  all 


multiplied  with  (  — )  . 

circuits  to  the  same  number  of  effective  turns  is  convenient  by 
eliminating  constant  factors  from  the  equations,  and  so  permit- 
ting a  direct  comparison.  .  When  speaking,  therefore,  in  the  fol- 
lowing of  the  impedance,  etc.,  of  the 
different  circuits,  we  always  refer  to 
their  reduced  values,  as  it  is  cus- 
tomary in  induction-motor  designing 
practice,  and  has  been  done  in  pre- 
ceding theoretical  investigations. 
173.  Let,  then,  in  Fig.  147: 
EQ)  /o,  ZQ  =  impressed  voltage, 
current  and  self-inductive  impedance 
respectively  of  a  stationary  circuit, 

EI,   /i,    Zi   =    impressed   voltage, 
current  and  self-inductive  impedance 
respectively  of  a  rotating  circuit, 

T  =  space  angle  between  the  axes  of  the  two  circuits, 
Z  =  mutual  inductive,  or  exciting  impedance  in  the  direction 
of  the  axis  of  the  stationary  coil, 

Z'  =  mutual  inductive,  or  exciting  impedance  in  the  direction 
of  the  axis  of  the  rotating  coil, 

Z"  =  mutual  inductive  or  exciting  impedance  in  the  direction 
at  right  angles  to  the  axis  of  the  rotating  coil, 


FIG.  147. 


ALTERNATING-CURRENT  MOTORS  305 

$  =  speed,  as  fraction  of  synchronism,  that  is,  ratio  of  fre- 
quency of  rotation  to  frequency  of  alternation. 

It  is  then : 

E.m.f.  consumed  by  self-inductive  impedance,  Zo/o- 

E.m.f.  consumed  by  mutual-inductive  impedance,  Z  (70  +  /i 
cos  r)  since  the  m.m.f.  acting  in  the  direction  of  the  axis  of  the 
stationary  coil  is  the  resultant  of  both  currents.  Hence: 

Eo  =  Z0/o  +  Z(/o  +  /iCOSr).  (3) 

In  the  rotating  circuit,  it  is: 

E.m.f.  consumed  by  self-inductive  impedance,  Zi/i. 

E.m.f.  consumed  by  mutual-inductive  impedance  or  "e.m.f.  of 
alternation":  Z'  (/i  +  70  cos  r).  (4) 

E.m.f.  of  rotation,  —jSZ"Jo  sin  r.  (5) 

Hence  the  impressed  e.m.f . : 

E,  =  Zi/i  +  Z'  (/!  +  70  cos  r)  -jSZ"!o  sin  r.  (6) 

In  a  structure  with  uniformly  distributed  winding,  as  used  in 
induction  motors,  etc.,  Z'  =  Z"  —  Z,  that  is,  the  exciting  im- 
pedance is  the  same  in  all  directions. 

Z  is  the  reciprocal  of  the  " exciting  admittance,"  Y  of  the  in- 
duction-motor theory. 

In  the  most  general  case,  of  a  motor  containing  n  circuits,  of 
which  some  are  revolving,  some  stationary,  if: 

Ek,  Ik,  Zk  =  impressed  e.m.f.,  current  and  self -inductive  im- 
pedance respectively  of  any  circuit,  k. 

Z*,  and  Z"  =  exciting  impedance  parallel  and  at  right  angles 
respectively  to  the  axis  of  a  circuit,  i, 

rkl  =  space  angle  between  the  axes  of  coils  k  and  i,  and 
S  =  speed,   as   fraction    of   synchronism,   or   "  frequency  of 
rotation." 

It  is  then,  in  a  coil,  i: 


n  n 


Ei  =  ZJi  +  Z*  }k  Ik  cos  T*<  -  jSZ"  )k  Ik  sin  rk\          (7) 

i  i 

where  : 

Zili  —  e.m.f.  of  self-inductive  impedance;  (8) 

n 

cos  Tfc*  =  e.m.f.  of  alternation;  (9) 

k  sin  TA*  =  e.m.f.  of  rotation;         (10) 


which  latter  =  0  in  a  stationary  coil,  in  which  S  =  0. 
20 


306  ELECTRICAL  APPARATUS 

The  power  output  of  the  motor  is  the  sum  of  the  powers  of  all 
the  e.m.fs.  of  rotation,  hence,  in  vector  denotation: 


=  -  S)^  tfZ"i*/*sin7V,  7J1,  (11) 

i  i 

and  herefrom  the  torque,  in  synchronous  watts: 

P  =  ^  =  ~  )I  Uzii5*lk  sin  rk\  Ii]1.  (12) 

o  11 

The  power  input,  in  vector  denotation,  is : 

n 


(13) 


and  therefore: 

Po1  =  true  power  input; 

P07  =  wattless  volt-ampere  input; 

V2  2 

Po1   +  Po7    =  apparent,    or   volt-ampere 

input ; 
p 
p-|  =  efficiency; 

Q  =  apparent  efficiency; 

^5-:  =  torque  efficiency; 
Po 

^.  —  apparent  torque  efficiency; 

Po1 

-Q    =  power-factor. 

From  the  n  circuits,  i  =  1,  2  .  .  .  n,  thus  result  n  linear 
equations,  with  2  n  complex  variables,  /;  and  #;. 

Hence  n  further  conditions  must  be  given  to  determine  the 
variables.  These  obviously  are  the  conditions  of  operation  of 
the  n  circuits. 

Impressed  e.m.fs.  E{  may  be  given. 

Or  circuits  closed  upon  themselves  Ei  =  0. 

Or  circuits  connected  in  parallel  CiEi  —  CkEk,  where  Ci  and  Ck 


ALTERNATING-CURRENT  MOTORS  307 

are  the  reduction  factors  of  the  circuits  to  equal  number  of 
effective  turns,  as  discussed  before. 

Or  circuits  connected  in  series:  ~-  —  --•>  etc. 

d       ck 

When  a  rotating  circuit  is  connected  through  a  commutator, 
the  frequency  of  the  current  in  this  circuit  obviously  is  the  same 
as  the  impressed  frequency.  Where,  however,  a  rotating  circuit 
is  permanently  closed  upon  itself,  its  frequency  may  differ  from 
the  impressed  frequency,  as,  for  instance,  in  the  polyphase  in- 
duction motor  it  is  the  frequency  of  slip,  s  =  1  —  S,  and  the 
self -inductive  reactance  of  the  circuit,  therefore,  is  sx ;  though  in 
its  reaction  upon  the  stationary  system  the  rotating  system  nec*- 
essarily  is  always  of  full  frequency. 

As  an  illustration  of  this  method,  its  application  to  the  theory 
of  some  motor  types  shall  be  considered,  especially  such  motors 
as  have  either  found  an  extended  industrial  application,  or  have 
at  least  been  seriously  considered. 

1.  POLYPHASE  INDUCTION  MOTOR 

174.  In  the  polyphase  induction  motor  a  number  of  primary 
circuits,  displaced  in  position  from  each  other,  are  excited  by 
polyphase  e.m.fs.  displaced  in  phase  from  each  other  by  a  phase 
angle  equal  to  the  position  angle  of  the  coils.  A  number  of  sec- 
ondary circuits  are  closed  upon  themselves.  The  primary  usu- 
ally is  the  stator,  the  secondary  the  rotor. 

In  this  case  the  secondary  system  always  offers  a  resultant 
closed  circuit  in  the  direction  of  the  axis  of  each  primary  coil, 
irrespective  of  its  position. 

Let  us  assume  two  primary  circuits  in  quadrature  as  simplest 
form,  and  the  secondary  system  reduced  to  the  same  number  of 
phases  and  the  same  number  of  turns  per  phase  as  the  primary 
system.  With  three  or  more  primary  phases  the  method  of 
procedure  and  the  resultant  equations  are  essentially  the  same. 

Let,  in  the  motor  shown  diagrammatically  in  Fig.  148: 

EQ  and  —  jEo,  70  and  —  jI0,  Z0  =  impressed  e.m.f.,  currents 
and  self-inductive  impedance  respectively  of  the  primary  system. 

0,  fi  and  —jjit  %i  =  impressed  e.m.f.,  currents  and  self-in- 
ductive impedance  respectively  of  the  secondary  system,  reduced 
to  the  primary.  Z  =  mutual-inductive  impedance  between 
primary  and  secondary,  constant  in  all  directions. 


308 


ELECTRICAL  APPARATUS 


S  —  speed;  s  =  1  —  S  =  slip,  as  fraction  of  synchronism. 
The  equation  of  the  primary  circuit  is  then,  by  (7) : 


J?     —    7   T     -L.    7  (  J  T  \ 

-v/o  —  "O/o  "i    "  \i  o  —  /  i/ • 

The  equation  of  the  secondary  circuit: 

0    =    Z!/!   +  Z  (7x    -    70)    +JSZ  (Jh    -  J4 

from  (15)  follows: 

T         T       Z° (1  ~  ^  ^         Zs 

-1       '°Z(1  -  S)  H-Zi       -°Zs  +  Zi 


FIG.  148. 


and,  substituted  in  (14) : 
Primary  current: 


Zs  +  Zl 


Zs 


ZZ0s 
Secondary  current: 

ZZoS 
Exciting  current: 

/oo   =  IQ  —  Ii  =  EQ  „„  — 

ZZoS  +  ZZi  + 

E.m.f.  of  rotation: 

#'  =  jsz  07i  -  j/o)  =  >sz  (/o  - 


=  (!-«) 


(14) 


(15) 


(16) 


(17) 


(18) 


(19) 


ZZqS  +  ZZi  + 


(20) 


ALTERNATING-CURRENT  MOTORS  309 

It  is,  at  synchronism;  s  =  0: 


7i   =0; 

/oo  =  /o 


•      ~~    17    I     17 

z  +  z»     1  + 

At  standstill: 

«  =  1;     B 


T 

" 


_ 

ZZo  +  ZZi  +  ZoZi' 


_          _ 

=  ZZo  4-  ZZ!  +  ZoZ/ 
E'  =  0. 

Introducing  as  parameter  the  counter  e.m.f.,  or  e.m.f.  of  mutual 
induction  : 

E  =  E0  -  Zo/o,  (21) 

or: 

EQ  =  E   +  Zo/o,  (22) 

it  is,  substituted  : 
Counter  e.m.f.  : 


'"ZZoS  +  ZZi- 

hence: 

Primary  impressed  e.m.f. : 

EQ  =  E  —         — ^^ — -         — '  (24) 

E.m.f.  of  rotation: 

E'  =  ES  =  E(l  -  «).  (25) 

Secondary  current: 

7i    =-f'-  (26) 

Primary  current: 

7.   =^??^1  =  li  +  f-  (27) 


310  ELECTRICAL  APPARATUS 

Exciting  current: 

/„„  =  J  =  EY.  (28) 

These  are  the  equations  from  which  the  transformer  theory  of 
the  polyphase  induction  motor  starts. 

175.  Since  the  frequency  of  the  secondary  currents  is  the  fre- 
quency of  slip,  hence  varies  with  the  speed,  S  =  1  —  s,  the  sec- 
ondary self -inductive  reactance  also  varies  with  the  speed,  and 
so  the  impedance: 

Zi  =  ri  +  jsxt.  (29) 

The  power  output  of  the  motor,  per  circuit,  is: 
P  =  IP,  h] 


[ZZ0s  +  ZZl 

where  the  brackets  [  ]  denote  the  absolute  value  of  the  term  in- 
cluded by  it,  and  the  small  letters,  e0,  z,  etc.,  the  absolute  values 
of  the  vectors,  E0,  Z,  etc. 

Since  the  imaginary  term  of  power  seems  to  have  no  physical 
meaning,  it  is  : 
Mechanical  power  output: 

e0Ws  (1  -  s) 


"  [ZZQs  +  ZZ,  + 

This  is  the  power  output  at  the  armature  conductors,  hence  in- 
cludes friction  and  windage. 
The  torque  of  the  motor  is  : 


1  -  s 

[ZZ0s  +~ZZi  +  ZoZ^P  ~  J  [ZZoS 


The  imaginary  component  of  torque  seems  to  represent  the 
radial  force  or  thrust  acting  between  stator  and  rotor.  Omitting 
this  we  have: 

0   2~  2  r   Q 
-j-^  t/Q    &        I  lo  /OO\ 

=  [ZZ0s+~ZZi+ZoZi]2' 


ALTERNATING-CURRENT  MOTORS  311 

The  power  input  of  the  motor  per  circuit  is  : 
P  o    =  [Eo,  /o] 


2  Zs  +  Z 


where: 

P'o  =  true  power, 

Po*  =  reactive  or  '/wattless  power," 

Q     =  \/P'o2  +  Po72  =  volt-ampere  input. 

Herefrom  follows  power-factor,  efficiency,  etc. 
Introducing  the  parameter:  #,  or  absolute  e,  we  have: 
Power  output: 

P  =  [F,  /.] 

-ha       '      --; 


Power  input: 

P-  \T?     T  1 
o  —  Lvo>  4 oj 

+  ZZi  +  ZQZi         Zs  + 


,  r 

=  e 


+  Zi)'  ,       Zs  +  Z 


r    o     g         i     ,   T 

L~  ~zzT          ' 


(35) 


'0  —  jXo)    ~\ 2  (fl  ~~  -7SXl)   ~*~   ^2   ^  ~  ^ 

=  ^'o2  (r0  -  jxo)  +  ^'i2  (^  -  j»i)  +  ^'oo2  (r  -  jx).  (36) 

And  since: 

-  =  -      -  ri  =  -  -  +  ri, 
s  s  s 


312  ELECTRICAL  APPARATUS 

and: 


it  is: 

f  o  =  Wro  +  z'lVi  +  too2/1  +  P)  -  j  Wteo  +  *'i2zi  +  *oo2z).    (37) 
Where  : 

iozr0  =  primary  resistance  loss, 

ii2ri  =  secondary  resistance  loss, 

ioQ2r  =  core  loss  (and  eddy-current  loss), 

P       =  output, 

io2x0  =  primary  reactive  volt-amperes, 

ii2£i  =  secondary  reactive  volt-amperes, 

ioo2x  =  magnetizing  volt-amperes. 

176.  Introducing  into  the  equations,  (16),  (17),  (18),  (19),  (23) 
the  terms: 


- 
"Z         o' 


(38) 


Where  Xo  and  Xi  are  small  quantities,  and  X  =  Xo  +  Xi  is  the 
"  characteristic  constant"  of  the  induction  motor  theory,  it  is: 
Primary  current: 

,       =   EQ  S  +  Xi  =  EQ     S  +  Xi  .       . 

'   Z    sXo  +  Xi  +  XoXi        Z    sXo  +  X 
Secondary  current: 

S  E*  S 


T        = 

' 


Z    sXo  +  Xi  +  X0Xi        Z    sXo  +  Xi 
Exciting  current: 


_ 


Z    sX0  +  Xi  +  X0Xi  ~~  Z 


,.   . 


E.m.f.  of  rotation: 


77T/  TTf      Q       _1  T7,      r,  A!  //IO\ 

-C/       =£1^0^     ~^T —    i_    \    '     [  x    x       ==    VO^      "^T i  .    \    '  V^-"/ 

S\o  ~r    AI  ~\  AoAi  SAo  ~i     AI 

Counter  e.m.f. : 

7^  rr  Xl  Xl 


\ _i_  \  \          - 

AI  +  XoXi  sXo  + 


ALTERNATING-CURRENT  MOTORS 


313 


177.  As  an  example  are  shown,  in  Fig.  149,  with  the  speed 
as  abscissae,  the  curves  of  a  polyphase  induction  motor  of  the 
constants : 

eQ  =  320  volts, 

Z  =  1  +  10y  ohms, 

Z0  =  Zl  =  0.1  +  0.3  j  ohms; 
hence: 

X0  =  Xi  =  0.0307  -  0.0069  j. 


P&D 
140 
130 
120 
110 
100 
90 
80 
70 
60 
50 
40 
30 
20 
10 
0 

1 

100 
90 
80 
70 
GO 
50 
40 
30 
20 
10 
0 

POLYPHASE  INDUCTION  MOTOR 
320  VOLTS 

x 

*~\ 

z 

N 

i 

y 

^\ 

\          I 



I 

x^ 

/ 

Vi 

"x^- 

L_^ 

/ 

^s 

' 

5 

x3r^ 

^^ 

/ 

N 

\   s^ 

x  \  1    40° 

\    1      350 

D 

^^ 

P 

/ 

X 

\ 

1    '        °0f) 

-- 

"^ 

~7 

^ 

y 

\ 

*           \   |       orrt 

p 

^ 

^ 

\         \  |       ^OU 

\     \  '     ?nn 

_—•—•—• 





•  —•-•"" 

y 

\   \ 

\        l]         1CA 

^^ 

^v 

^** 

^ 

*^"^ 

^  

<sst>^ 

0.1         0.2        0.3         0.4         0.5         0.6         0.7         0.8         0.9         1.0 

It  is: 


FIG.  149. 


320 {10.30s  -  (s  +  0.1)  j} 
(L03  +  1.63  s)  -  j  (0.11  -  5.99^)  amp' 
2048(1  -  s) 


P  =  (1  -  s)  D 

0.11  -  5.99s 


tan  0"  = 


1.03  +  1.63 


cos  (0r  -  0")  =  power-factor. 

Fig.  149  gives,  with  the  speed  S  as  abscissae:  the  current,  /; 
the  power  output,  P;  the  torque,  D;  the  power-factor,  p;  the 
efficiency,  77. 


314  ELECTRICAL  APPARATUS 

The  curves  show  the  well-known  characteristics  of  the  poly- 
phase induction  motor:  approximate  constancy  of  speed  at  all 
loads,  and  good  efficiency  and  power-factor  within  this  narrow- 
speed  range,  but  poor  constants  at  all  other  speeds. 

1.  SINGLE-PHASE  INDUCTION   MOTOR 

178.  In  the  single-phase  induction  motor  one  primary  circuit 
acts  upon  a  system  of  closed  secondary  circuits  which  are  dis- 
placed from  each  other  in  position  on  the  secondary  member. 

Let  the  secondary  be  assumed  as  two-phase,  that  is,  containing 
or  reduced  to  two  circuits  closed  upon  themselves  at  right  angles 


FIG.  150. — Single-phase  induction  motor. 

to  each  other.  While  it  then  offers  a  resultant  closed  secondary 
circuit  to  the  primary  circuit  in  any  position,  the  electrical  dis- 
position of  the  secondary  is  not  symmetrical,  but  the  directions 
parallel  with  the  primary  circuit  and  at  right  angles  thereto  are 
to  be  distinguished.  The  former  may  be  called  the  secondary 
energy  circuit,  the  latter  the  secondary  magnetizing  circuit,  since 
in  the  former  direction  power  is  transferred  from  the  primary  to 
the  secondary  circuit,  while  in  the  latter  direction  the  secondary 
circuit  can  act  magnetizing  only. 

Let,  in  the  diagram  Fig.  150: 

EQ,  /o,  Z0  =  impressed  e.m.f.,  current  and  self -inductive  im- 
pedance, respectively,  of  the  primary  circuit, 

71,  Zi  =  current  and  self-inductive  impedance,  respectively, 
of  the  secondary  energy  circuit, 

72,  Zi  =  current  and  self-inductive  impedance,  respectively, 
of  the  secondary  magnetizing  circuit, 

Z  =  mutual-inductive  impedance, 
S  =  speed, 

and  let  s0  =  1  —  S2  (where  s0  is  not  the  slip) . 
It  is  then,  by  equation  (7) : 


ALTERNATING-CURRENT  MOTORS  315 

Primary  circuit: 

Eo  =  Zo/o  +  Z(10-  /i).  (44) 

Secondary  energy  circuit: 

0  =  Z!/!  +  Z  (/!  -  7o)  -  JSZI*  (45) 

Secondary  magnetizing  circuit  : 

0  =  Zi/8  +  ZI,  -  JSZ  (/o  -  /i)  ;  (46) 

hence,  from  (45)  and  (46)  : 

.  r        Z  (Zs0  +  Zi) 


It  is,  at  synchronism,  >S  =  1,  s0  =  0: 

r        F  2Z  + 

•° 


77 


and,  substituted  in  (44) : 
Primary  current: 

70  =  EO  -  (49) 

Secondary  energy  current: 


/,  =  £„-      ^  (50) 
Secondary  magnetizing  current: 

/j  /j  1  /  I*  -t  \ 

TF1'  (51) 


E.m.f.  of  rotation  of  secondary  energy  circuit: 

77 

fr  =  -  jSZ/,_=  S'&   -£•  (52) 

E.m.f.  of  rotation  of  secondary  magnetizing  circuit: 

$'1  =  -  JSZ  (/„  -/,)  =  -  JSTS,  ZZl(^+Zl);        (53) 

where  : 

K  =  Z0  (Z2s0  +  2  ZZi  +  Zi2)  +  ZZ!  (Z  +  Zi).          (54) 


•° 


316  ELECTRICAL  APPARATUS 

Hence,  at  synchronism,  the  secondary  current  of  the  single- 
phase  induction  motor  does  not  become  zero,  as  in  the  polyphase 
motor,  but  both  components  of  secondary  current  become  equal. 

At  standstill,  S  =  0,  SQ  =  1,  it  is: 

•f  0  V  0    ~rr  r/       1       vTz        i       >7    rr    } 


•     ZZ0  +  ZZl 

h  =  o. 

That  is,  primary  and  secondary  current  corresponding  thereto 
have  the  same  values  as  in  the  polyphase  induction  motor,  as 
was  to  be  expected. 

179.  Introducing  as  parameter  the  counter  e.m.f.,  or  e.m.f.  of 
mutual  induction: 

E    =    EQ   —   Zo/0, 

and  substituting  for  70  from  (49),  it  is: 
Primary  impressed  e.m.f.: 

Zo(Z*8o  +  2ZZi  +  Zx2)  +  ZZ,  (Z  +  Z,)         ,     , 

*•"*"          -zzrcz  +  zo 

Primary  current: 

T         J-1Z*s0  +  2ZZl  +  Zl*  ,     . 

h  =  $  "  "zz^z  +  ZiT 

Secondary  energy  circuit: 

^      ZSQ  +  Zi 

•  = 


Secondary  magnetizing  circuit: 

/-+^fzr'  • 

(GO) 


And: 

(61) 


These  equations  differ  from  the  equations  of  the  polyphase 

induction  motor  by  containing  the  term  s0  =  (1  —  S2),  instead 

Sfj5* 
of  s  =  (1  —  S),  and  by  the  appearance  of  the  terms,  ~  .    „   and 

of  frequency  (1  +  3),  in  the  secondary  circuit. 


ALTERNATING-CURRENT  MOTORS  317 

The  power  output  of  the  motor  is: 

P  =  [#i,  /i]  +  [E2,  h] 

,i,  Zs0  +  ZJ  -  [Zi  (Z  +  ZO,  ZJ} 
(soz2  -  Z!2) 


m 

and  the  torque,  in  synchronous  watts: 


(62) 


P       ££02z2ri  (s032  -  Z!2)  . 

D  ==  -g   --  -     rgp" 

From  these  equations  it  follows  that  at  synchronism  tor- 
que and  power  of  the  single-phase  induction  motor  are  already 
negative. 

Torque  and  power  become  zero  for: 

s0z2  —  Zi2  =  o, 
hence : 

(64) 

that  is,  very  slightly  below  synchronism. 

Let  z  =  10,  zi  =  0.316,  it  is,  S  =  0.9995. 

In  the  single-phase  induction  motor,  the  torque  contains  the 
speed  S  as  factor,  and  thus  becomes  zero  at  standstill. 

Neglecting  quantities  of  secondary  order,  it  is,  approximately: 

'         "  ZS°~Zl)+2Z»Zl'  ||      (65) 

ZS°  +  Zl  ,  (66) 


2 


(68) 


77 


.  ( 


ZO  +2Z0Z!]2 

This  theory  of  the  single-phase  induction  motor  differs  from 
that  based  on  the  transformer  feature  of  the  motor,  in  that  it 
represents  more  exactly  the  phenomena  taking  place  at  inter- 


318 


ELECTRICAL  APPARATUS 


mediate  speeds,  which  are  only  approximated  by  the  transformer 
theory  of  the  single-phase  induction  motor. 

For  studying  the  action  of  the  motor  at  intermediate  and  at 
low  speed,  as  for  instance,  when  investigating  the  performance 
of  a  starting  device,  in  bringing  the  motor  up  to  speed,  that  is, 
during  acceleration,  this  method  so  is  more  suited.  An  applica- 
tion to  the  "  condenser  motor,"  that  is,  a  single-phase  induction 
motor  using  a  condenser  in  a  stationary  tertiary  circuit  (under 
an  angle,  usually  60°,  with  the  primary  circuit)  is  given  in  the 
paper  on  "  Alternating-Current  Motors,"  A.  I.  E.  E.  Transac- 
tions, 1904. 


P&D 

120 

110 

100 

90 


SINGLE  PHASE 

NDUCTION  MOTOR 

400  VOLTS 


100 
90 
80 
70 
60 
50 
40 
30 
20 
10 


.4  .5          .6 

FIG.  151. 

180.  As  example  are  shown,  in  Fig.  151,  with  the  speed  as 
abscissae,  the  curves  of  a  single-phase  induction  motor,  having 
the  constants: 

e0  =  400  volts, 

Z  =  1  +  10  j  ohms, 


I 

-700— 
-650— 


600— 
550— 
-500— 
450— 
100— 


and: 
hence : 


Z0  =  Zi  =  0.1  4-  0.3  j  ohms; 

IQ  =  400  ^  amp.; 

N  =  (s0  +  0.2)  4- j. (10  s0  4-  0.6  -  0.6  S); 

K  =  (0.1+  0.3 j)N+(l  +  10j)(0.1+j)  (0.3-0.3 S)- 


n       1616  Ss0 

~  *~  syncnronous 


ALTERNATING-CURRENT  MOTORS 


319 


Fig.  151  gives,  with  the  speed,  S,  as  abscissae:  the  current,  7o, 
the  power  output,  P,  the  torque,  D,  the  power-factor,  p,  the 
efficiency,  r;. 

3.  POLYPHASE  SHUNT  MOTOR 

181.  Since  the  characteristics  of  the  polyphase  motor  do  not 
depend  upon  the  number  of  phases,  here,  as  in  the  preceding,  a 
two-phase  system  may  be  assumed:  a  two-phase  stator  winding 
acting  upon  a  two-phase  rotor  winding,  that  is,  a  closed-coil 
rotor  winding  connected  to  the  commutator  in  the  same  manner 
as  in  direct-current  machines,  but  with  two  sets  of  brushes  in 
quadrature  position  excited  by  a  two-phase  system  of  the  same 
frequency.  Mechanically  the  three-phase  system  here  has  the 
advantage  of  requiring  only  three  sets  of  brushes  instead  of  four 


FIG.  152. 

as  with  the  two-phase  system,  but  otherwise  the  general  form 
of  the  equations  and  conclusions  are  not  different. 

Let  EQ  and  —  j#o  =  e.m.fs.  impressed  upon  the  stator,  #1  and 
—  jEi  =  e.m.fs.  impressed  upon  the  rotor,  00  =  phase  angle  be- 
tween e.m.f.,  E0  and  EI,  and  6\  =  position  angle  between  the 
stator  and  rotor  circuits.  The  e.m.fs.,  EQ  and  —jEo,  produce  the 
same  rotating  e.m.f.  as  two  e.m.fs.  of  equal  intensity,  but  dis- 
placed in  phase  and  in  position  by  angle  00  from  E0  and  jEQ, 
and  instead  of  considering  a  displacement  of  phase,  0o,  and  a  dis- 
placement of  position,  0\,  between  stator  and  rotor  circuits,  we 
can,  therefore,  assume  zero-phase  displacement  and  displacement 
in  position  by  angle  00  +  0i  =  0.  Phase  displacement  between 
stator  and  rotor  e.m.fs.  is,  therefore,  equivalent  to  a  shift  of 
brushes,  hence  gives  no  additional  feature  beyond  those  pro- 
duced by  a  shift  of  the  commutator  brushes. 


320  ELECTRICAL  APPARATUS 

Without  losing  in  generality  of  the  problem,  we  can,  therefore, 
assume  the  stator  e.m.fs.  in  phase  with  the  rotor  e.m.fs.,  and  the 
polyphase  shunt  motor  can  thus  be  represented  diagrammatically 
by  Fig.  152. 

182.  Let,  in  the  polyphase  shunt  motor,  shown  two-phase  in 
diagram,  Fig.  152: 

EQ  and  —jE0,  I0  and  —  j/o,  Z0  =  impressed  e.m.fs.,  currents 
and  self-inductive  impedance  respectively  of  the  stator  circuits, 

cE0  and  —jcE0)  /i  and  —  j/i,  Zi  =  impressed  e.m.fs.,  currents 
and  self-inductive  impedance  respectively  of  the  rotor  circuits, 
reduced  to  the  stator  circuits  by  the  ratio  of  effective  turns,  c, 

Z  =  mutual-inductive  impedance, 

S  =  speed;  hence  s  =  1  —  S  =  slip, 

0  =  position  angle  between  stator  and  rotor  circuits,  or 
"  brush  angle." 

It  is  then : 
Stator: 

E\  =  Zo/o  +  Z  (/o  -  /i  cos  0  -  j/!  sin  0).  (72) 

Rotor : 

cEQ  =  Zi/i  +  Z  (/i  -  /o  cos  0  +  jlo  sin  0)  - 

jSZ  (  -  j/i  +  /o  sin  0  +  JJ0  cos  0).     (73) 
Substituting : 

a  ==  cos  0  —  j  sin  0, 
5  =  cos  0  -f  j  sin  0, 
it  is: 

(76  =  1,  (75) 

and: 

EQ  =  Zo/o  +  Z  (/o  -  Vi),  (76) 

cEQ  =  Z,h  +  Z(h~  <r/o)  +  JSZ  (jh  ~  Wo) 

=  Zi/!  +  sZ  (/!  -  <r/o).  (77) 

Herefrom  follows: 

/O    =    EQ      „„        ,       ^r^        I       ^    r,   >  ('8) 


(74) 


+  c)  Z  +  cZi  ,     , 

(79) 


for  c  =  o,  this  gives: 


ALTERNATING-CURRENT  MOTORS  321 

that  is,  the  polyphase  induction-motor  equations,  a  =  cos  6  + 

^ 

j  sin  0  =  I*-  representing  the  displacement  of  position  between 
stator  and  rotor  currents. 

This  shows  the  polyphase  induction  motor  as  a  special  case  of 
the  polyphase  shunt  motor,  for  c  =  of 

The  e.m.fs.  of  rotation  are: 

E"i  =  -JSZ  (-jfi  +  /o  sin  0  +  j/o  cos  6) 

=  SZ(<r/o-/i); 
hence: 


V          QJ?  —  CZ0)  ,R    . 

^'  ' 


The  power  output  of  the  motor  is  : 
=  [Ei,  7J 

~  cZo}  Z> 


which,  suppressing  terms  of  secondary  order,  gives: 

mO  —  r0cos  0))+c(riCos  0+#isin0  —  cr0)} 


o  +  ZZi  + 

(82) 
for  iSc  =  o,  this  gives: 


~  [sZZQ  +  ZZ, 

the  same  value  as  for  the  polyphase  induction  motor. 

In  general,  the  power  output,  as  given  by  equation  (82),  be- 
comes zero: 

P  =  o, 

for  the  slip: 

fi  cos  6  +  Xi  sin  6  —  cr0  , 

SQ    =     —C     --  :  -  ~,  -  :  -  —  —  rr'  (83) 

ri  +  c  (x0  sin  0  —  r0  cos  0) 

183.  It  follows  herefrom,  that  the  speed  of  the  polyphase 
shunt  motor  is  limited  to  a  definite  value,  just  as  that  of  a  direct- 
current  shunt  motor,  or  alternating-current  induction  motor. 
In  other  words,  the  polyphase  shunt  motor  is  a  constant-speed 
motor,  approaching  with  decreasing  load,  and  reaching  at  no- 
load  a  definite  speed: 

S0  =  1  -  s0.  (84) 

The  no-load  speed,  SQ,  of  the  polyphase  shunt  motor  is,  how- 
ever, in  general  not  synchronous  speed,  as  that  of  the  induction 

21 


322  ELECTRICAL  APPARATUS 

motor,  but  depends  upon  the  brush  angle,  6,  and  the  ratio,  c,  of 
rotor  -T-  stator  impressed  voltage. 

At  this  no-load  speed,  SQ,  the  armature  current,  /i,  of  the 
polyphase  shunt  motor  is  in  general  not  equal  to  zero,  as  it  is 
in  the  polyphase  induction  motor. 

Two  cases  are  therefore  of  special  interest  : 

1.  Armature  current,  7i  =  o,  at  no-load,  that  is,  at  slip,  SQ. 

2.  No-load  speed  equals  synchronism,  s0  =  o,. 
1.  The  armature  or  rotor  current  (79): 

asZ  +  c(Z  +  Z,) 
'  '  °  sZZQ  +  ZZl  +  ZQZi 
becomes  zero,  if: 


or,  since  Zi  is  small  compared  with  Z}  approximately: 

c  =  —ffs  =  —s  (cos  6  —  j  sin  0); 
hence,  resolved: 

c  =  —s  cos  0, 

o  =  s  sin  0; 
hence  : 

0  =  o 


(85) 

That  is,  the  rotor  current  can  become  zero  only  if  the  brushes 
are  set  in  line  with  the  stator  circuit  or  without  shift,  and  in  this 
case  the  rotor  current,  and  therewith  the  output  of  the  motor, 
becomes  zero  at  the  slip,  s  =  —  c. 

Hence  such  a  motor  gives  a  characteristic  curve  very  similar 
to  that  of  the  polyphase  induction  motor,  except  that  the  stator 
tends  not  toward  synchronism  but  toward  a  definite  speed  equal 
to  (1  +  c)  times  synchronism. 

The  speed  of  such  a  polyphase  motor  with  commutator  can, 
therefore,  be  varied  from  synchronism  by  the  insertion  of  an 
e.m.f.  in  the  rotor  circuit,  and  the  percentage  of  variation  is  the 
same  as  the  ratio  of  the  impressed  rotor  e.m.f.  to  the  impressed 
stator  e.m.f.  A  rotor  e.m.f.,  in  opposition  to  the  stator  e.m.f. 
reduces,  in  phase  with  the  stator  e.m.f.,  increases  the  free-run- 
ning speed  of  the  motor.  In  the  former  case  the  rotor  impressed 
e.m.f.  is  in  opposition  to  the  rotor  current,  that  is,  the  rotor 
returns  power  to  the  system  in  the  proportion  in  which  the  speed 


ALTERNATING-CURRENT  MOTORS  323 

is  reduced,  and  the  speed  variation,  therefore,  occurs  without 
loss  of  efficiency,  and  is  similar  in  its  character  to  the  speed  con- 
trol of  a  direct-current  shunt  motor  by  varying  the  ratio  between 
the  e.m.f  .  impressed  upon  the  armature  and  that  impressed  upon 
the  field. 

Substituting  in  the  equations  : 


it  is: 

Z1  _ 


sZZo  +  ZZ 


sZZ 


i       ZoZi'  (88) 

P  -  -  cr0) 

+  ZZ!  +  ZiZi}* 


These  equations  of  70  and  /i  are  the  same  as  the  polyphase 
induction-motor  equations,  except  that  the  slip  from  synchron- 
ism, s,  of  the  induction  motor,  is,  in  the  numerator,  replaced  by 
the  slip  from  the  no-load  speed,  SL 

Insertion  of  voltages  into  the  armature  of  an  induction  motor 
in  phase  with  the  primary  impressed  voltages,  and  by  a  com- 
mutator, so  gives  a  speed  control  of  the  induction  motor  without 
sacrifice  of  efficiency,  with  a  sacrifice,  however,  of  the  power- 
factor,  as  can  be  shown  from  equation  (87). 

184.  2.  The  no-load  speed  of  the  polyphase  shunt  motor  is  in 
synchronism,  that  is,  the  no-load  slip,  s0  =  o,  or  the  motor  out- 
put becomes  zero  at  synchronism,  just  as  the  ordinary  induction 
motor,  if,  in  equation  (83)  : 

ri  cos  0  -f-  Xi  sin  6  —  crQ  =  o; 
hence: 


C    = 


TO 
or,  substituting: 

^  =  tan  «i,  (91) 

where  on  is  the  phase  angle  of  the  rotor  impedance,  it  is: 
c  =  —  cos  (ai  —  6), 

7*0 


324  ELECTRICAL  APPARATUS 

or: 

cos  («i  -  0)  =  p  c,  (92) 

or: 

c  =  "  COS  (a'  -  9).  (93) 

* 


Since  r0  is  usually  very  much  smaller  than  z\,  if  c  is  not  very 
large,  it  is  : 

cos  («i  —  6)  =  o; 
hence  : 

6  =  90°  -  «i.  (94) 

That  is,  if  the  brush  angle,  9,  is  complementary  to  the  phase 
angle  of  the  self-inductive  rotor  impedance,  «i,  the  motor  tends 
toward  approximate  synchronism  at  no-load. 

Hence  : 

At  given  brush  angle,  0,  a  value  of  secondary  impressed  e.m.f., 
cEo,  exists,  which  makes  the  motor  tend  to  synchronize  at  no- 
load  (93),  and, 

At  given  rotor-impressed  e.m.f.,  cE0,  a  brush  angle,  6,  exists, 
which  makes  the  motor  synchronize  at  no-load  (92). 

185.  3.  In  the  general  equations  of  the  polyphase  shunt  motor, 
the  stator  current,  equation  (78)  : 

sZ  +  Zi  +  dcZ 
'  '  °  sZZQ  +  ZZ, 

can  be  resolved  into  a  component: 

jn  TJ,    _  s%  +  Zi  _  , 

:  *°  ZZ1 


which  does  not  contain  c,  and  is  the  same  value  as  the  primary 
current  of  the  polyphase  induction  motor,  and  a  component: 


sZZ,,  +  ZZt  +  Z0Z 
Resolving  /"o,  it  assumes  the  form: 

/"o  =  E^c(A^  -jA2) 

=  c  [Ai  cos  (9  +  ^2  sin  0)  +  j  (Ai  sin  ^  -  A2  cos  ^)  }.    (97) 

This  second  component  of  primary  current,  7"0,  which  is  pro- 
duced by  the  insertion  of  the  voltage,  cE,  into  the  secondary  cir- 
cuit, so  contains  a  power  component: 

i'o  =  c  (Ai  cos  0  +  A2  sin  0),  (98) 


ALTERNATING-CURRENT  MOTORS  325 

and  a  wattless  or  reactive  component: 

i"Q  =  +jc  (Ai  sin  0  -  A2  cos  0);  (99) 

where : 

jit    ._  „•'        iV  nnrh 

4    o  —  *•  o  —  jf'   o»  v-'-*-'^'/ 

The  reactive  component,  t"o,  is  zero,  if: 

AI  sin  0  —  A2  cos  6  =  0;  (I0l) 

hence: 

tan  0i  =  +  ~-  (102) 

In  this  case,  that  is,  with  brush  angle,  0i,  the  secondary  im- 
pressed voltage,  cE,  does  not  change  the  reactive  current,  but 
adds  or  subtracts,  depending  on  the  sign  of  c,  energy,  and  so 
raises  or  lowers  the  speed  of  the  motor:  case  (1). 

The  power  component,  i'0,  is  zero,  if: 

Ai  cos  0  +  A2  sin  0  =  o,  (103) 

hence : 

tan  02  =  -  (104) 

In  this  case,  that  is,  with  brush  angle,  02,  the  secondary  im- 
pressed voltage,  cE,  does  not  change  power  or  speed,  but  pro- 
duces wattless  lagging  or  leading  current.  That  is,  with  the 
brush  position,  02,  the  polyphase  shunt  motor  can  be  made  to 
produce  lagging  or  leading  currents,  by  varying  the  voltage  im- 
pressed upon  the  secondary,  cE,  just  as  a  synchronous  motor 
can  be  made  to  produce  lagging  or  leading  currents  by  varying 
its  field  excitation,  and  plotting  the  stator  current,  /o,  of  such  a 
polyphase  shunt  motor,  gives  the  same  V-shaped  phase  charac- 
teristics as  known  for  the  synchronous  motor. 

These  two  phase  angles  or  brush  positions,  0i  and  02,  are  in 
quadrature  with  each  other. 

There  result  then  two  distinct  phenomena  from  the  insertion 
of  a  voltage  by  commutator,  into  an  induction-motor  armature: 
a  change  of  speed,  in  the  brush  position,  0i,  and  a  change  of  phase 
angle,  in  the  brush  position,  02,  at  right  angles  to  0i. 

For  any  intermediate  brush  position,  0,  a  change  of  speed  so 
results  corresponding  to  a  voltage : 

cE  cos  (0i  -  0) ; 


326  ELECTRICAL  APPARATUS 

and  a  change  of  phase  angle  corresponding  to  a  voltage: 

C#COS  (02   -    0), 

=  cE  sin  (0i  -  0), 

and  by  choosing  then  such  a  position,  0,  that  the  wattless  current 
produced  by  the  component  in  phase  with  02,  is  equal  and  op- 
posite to  the  wattless  lagging  current  of  the  motor  proper,  /'0, 
the  polyphase  shunt  motor  can  be  made  to  operate  at  unity 
power-factor  at  all  speeds  (except  very  low  speeds)  and  loads. 
This,  however,  requires  shifting  the  brushes  with  every  change 
of  load  or  speed. 

When  using  the  polyphase  shunt  motor  as  generator  of  watt- 
less current,  that  is,  at  no-load  and  with  brush  position,  02,  it  is : 

s  =  0; 
hence,  from  (78) : 


/'o  =  ^qr^  (106) 

or,  approximately: 

...  .;      :        /'.  =  §•• 

that  is,  primary  exciting  current: 

/"o  =  Eo  Zi  (z°+  Zoy  (107) 

or,  approximately,  neglecting  ZQ  against  Z: 

Trt     _  E0dc 
/    o  —  —fj— 


-  EQC  (cos  ^  +  J  s^n  0) 

7*1  +  j#i 

=  — ^  { (ri  cos  0  +  Xi  sin  0)  —  j  (a^x  cos  0  —  n  sin  0) } , 


(108) 


and,  since  the  power  component  vanishes: 

ri  cos  0  +  #1  sin  0  =  0, 
or: 

tan  02  =  --•  (109) 


ALTERNATING-CURRENT  MOTORS 
Substituting  (109)  in  (108)  gives: 

i  cos  02  -  n  sin  02) 


327 


/"0  =  - 


and: 


Zi 

/o  =  -ff  —  3 


(110) 


(in) 


186.  In  the  exact  predetermination  of  the  characteristics  of 
such  a  motor,  the  effect  of  the  short-circuit  current  under  the 
brushes  has  to  be  taken  into  consideration,  however.     When  a 
commutator  is  used,  by  the  passage  of  the  brushes  from  segment 
to  segment  coils  are  short-circuited.     Therefore,  in  addition  to 
the  circuits  considered  above,  a  closed  circuit  on  the  rotor  has 
to  be  introduced  in  the  equations  for  every  set  of  brushes.     Re- 
duced to  the  stator  circuit  by  the  ratio  of  turns,  the  self -inductive 
impedance  of  the  short-circuit  under  the  brushes  is  very  high, 
the  current,  therefore,  small,  but  still  sufficient  to  noticeably  af- 
fect the  motor  characteristics,  at  least  at  certain  speeds.     Since, 
however,  this  phenomenon  will  be  considered  in  the  chapters  on 
the  single-phase  motors,  it  may  be  omitted  here. 

4.  POLYPHASE  SERIES  MOTOR 

187.  If  in  a  polyphase  commutator  motor  the  rotor  circuits 
are  connected  in  series  to  the  stator  circuits,  entirely  different 

Io 


FIG.  153. 

characteristics  result,  and  the  motor  no  more  tends  to  synchronize 
nor  approaches  a  definite  speed  at  no-load,  as  a  shunt  motor,  but 
with  decreasing  load  the  speed  increases  indefinitely.  In  short, 


328  ELECTRICAL  APPARATUS 

the  motor  has  similar  characteristics  as  the  direct-current  series 
motor. 

In  this  case  we  may  assume  the  stator  reduced  to  the  rotor  by 
the  ratio  of  effective  turns. 

Let  then,  in  the  motor  shown  diagrammatically  in  Fig.  153: 

.Z?o  and  —JE0,  I0  and  —  jIQ,  Z0  =  impressed  e.m.fs.,  currents 
and  self-inductive  impedance  of  stator  circuits,  assumed  as  two- 
phase,  and  reduced  to  the  rotor  circuits  by  the  ratio  of  effective 
turns,  c, 

EI  and  —jEi,  7i,  and  —  jli,  Zi  =  impressed  e.m.fs.  currents 
and  self-inductive  impedance  of  rotor  circuits, 

Z  =  mutual-inductance  impedance, 

5  =  speed;  and,  s  =  1  —  S  =  slip, 

6  =  brush  angle, 

c   =  ratio  of  effective  stator  turns  to  rotor  turns. 

If,  then: 

E  and  —jE  =  impressed  e.m.fs.,  /  and  —  jl  =  currents  of 
motor,  it  is  : 

7i  =  7,  (112) 

7o  =  cl,  (113) 

cE0  +  #x  =  E'}  (114) 

and,  stator,  by  equation  (7)  : 

EQ  =  Zo/o  +  Z(Jo  -  1  1  cos  B  -  j7i  sin  0);  (115) 

rotor: 

El  =  ZJ,  +  Z  (/!  -  /o  cos  e  +  j/o  sin  6)  -  jSZ  (-  jh  +  /o 

sin  B  -f  j/ocos  6)',       (116) 

and,  e.m.f.  of  rotation: 

E\  =  -  JSZ  (-  jh  +  7o  sin  B  +  J7i  cos  e).  (117) 

Substituting  (Il2),  (113)  in  (115),  (116),  (117),  and  (115),  (116) 
in  (114)  gives: 

Tjl 

1  "  (c2Z0  +  Zi)  +  Z  (1  +  c2  -  2  c  cos  e)  +  SZ  (ca  -^T)5  (' 

where  : 

<T  =  cos  0  -  jsin  B,  (119) 

and: 

SZE  (<*-!)  . 


w, 
• 


_  _ 
(c2Z0  +  Zi)  =  Z  (I  +  c2  -  2ccos0)+SZ(cer-l)]' 

(120) 


ALTERNATING-CURRENT  MOTORS 


329 


and  the  power  output: 

P  =  U?'i,  /il' 

Se2  {  c  (r  cos  6  +  x  sin  0)  —  r 


[(c2Z0  +  Zi)  +  Z  (1  +  c2  -  2  c  cos  0)  +  SZ  (co-  -  I)]2 

(121) 

The  characteristics  of  this  motor  entirely  vary  with  a  change 

Se2r(x— 


of  the  brush  angle,  0.     It  is,  for  0  =  0:  P  = 


r 


hence 


POLYPHASE  SERIES  MOTOR 
64O  VOLTS 


.0         1.2        1.4 

FIG.  154. 


2.0 


very  small,  while  for  0  =  90°:  P  = 


—  r) 


,  hence  consider- 


able. Some  brush  angles  give  positive  P:  motor,  others  negative, 
P,  generator. 

In  such  a  motor,  by  choosing  0  and  c  appropriately,  unity 
power-factor  or  leading  current  as  well  as  lagging  current  can  be 
produced. 

That  is,  by  varying  c  and  0,  the  power  output  and  therefore 
the  speed,  as  well  as  the  phase  angle  of  the  supply  current  or 
the  power-factor  can  be  varied,  and  the  machine  used  to  produce 
lagging  as  well  as  leading  current,  similarly  as  the  polyphase 
shunt  motor  or  the  synchronous  motor.  Or,  the  motor  can  be 
operated  at  constant  unity  power-factor  at  all  loads  and  speeds 
(except  very  low  speeds),  but  in  this  case  requires  changing  the 


330  ELECTRICAL  APPARATUS 

brush  angle,  6,  and  the  ratio,  c,  with  the  change  of  load  and  speed. 
Such  a  change  of  the  ratio,  c,  of  rotor  -f-  stator  turns  can  be  pro- 
duced by  feeding  the  rotor  (or  stator)  through  a  transformer  of 
variable  ratio  of  transformation,  connected  with  its  primary  cir- 
cuit in  series  to  the  stator  (or  rotor). 

188.  As  example  is  shown  in  Fig.  154,  with  the  speed  as 
abscissa,  and  values  from  standstill  to  over  double  synchronous 
speed,  the  characteristic  curves  of  a  polyphase  series  motor  of 
the  constants: 

e  =  640  volts, 
Z  =  1  +  10  j  ohms, 
Z0  =  Z1  =  0.1  +  0.3  j  ohms, 
c  =  1, 

8  =  37°;  (sin  6  =  0.6;  cos  0  =  0.8); 
hence: 

T  =  640 

(0.6  +  5.8  S)  +  j  (4.6  -  2.6  S)  a 
4673  S 

K\fJ 


(0.6  +  5.8  S)2  +  (4.6  -  2.6  S) 

As  seen,  the  motor  characteristics  are  similar  to  those  of  the 
direct-current  series  motor:  very  high  torque  in  starting  and  at 
low  speed,  and  a  speed  which  increases  indefinitely  with  the  de- 
crease of  load.  That  is,  the  curves  are  entirely  different  from 
those  of  the  induction  motors  shown  in  the  preceding.  The 
power-factor  is  very  high,  much  higher  than  in  induction  motors, 
and  becomes  unity  at  the  speed  S  =  1.77,  or  about  one  and  three- 
quarter  synchronous  speed. 


CHAPTER  XX 

SINGLE-PHASE  COMMUTATOR  MOTORS 
I.  General 

189.  Alternating-current  commutating  machines  have  so  far 
become  of  industrial  importance  mainly  as  motors  of  the  series 
or  varying-speed  type,  for  single-phase  railroading,  and  as  con- 
stant-speed motors  or  adjustable-speed  motors,  where  efficient 
acceleration  under  heavy  torque  is  necessary.  As  generators, 
they  would  be  of  advantage  for  the  generation  of  very  low  fre- 
quency, since  in  this  case  synchronous  machines  are  uneconom- 
ical, due  to  their  very  low  speed,  resultant  from  the  low  frequency. 

The  direction  of  rotation  of  a  direct-current  motor,  whether 
shunt  or  series  motor,  remains  the  same  at  a  reversal  of  the  im- 
pressed e.m.f.,  as  in  this  case  the  current  in  the  armature  circuit 
and  the  current  in  the  field  circuit  and  so  the  field  magnetism 
both  reverse.  Theoretically,  a  direct-current  motor  therefore 
could  be  operated  on  an  alternating  impressed  e.m.f.  provided 
that  the  magnetic  circuit  of  the  motor  is  laminated,  so  as  to  fol- 
low the  alternations  of  magnetism  without  serious  loss  of  power, 
and  that  precautions  are  taken  to  have  the  field  reverse  simul- 
taneously with  the  armature.  If  the  reversal  of  field  magnetism 
should  occur  later  than  the  reversal  of  armature  current,  during 
the  time  after  the  armature  current  has  reversed,  but  before  the 
field  has  reversed,  the  motor  torque  would  be  in  opposite  direc- 
tion and  thus  subtract;  that  is,  the  field  magnetism  of  the  alter- 
nating-current motor  must  be  in  phase  with  the  armature  cur- 
rent, or  nearly  so.  This  is  inherently  the  case  with  the  series 
type  of  motor,  in  which  the  same  current  traverses  field  coils 
and  armature  windings. 

Since  in  the  alternating-current  transformer  the  primary  and 
secondary  currents  and  the  primary  voltage  and  the  secondary 
voltage  are  proportional  to  each  other,  the  different  circuits  of 
the  alternating-current  commutator  motor  may  be  connected 
with  each  other  directly  (in  shunt  or  in  series,  according  to  the 
type  of  the  motor)  or  inductively,  with  the  interposition  of  a 

331 


332  ELECTRICAL  APPARATUS 

transformer,  and  for  this  purpose  either  a  separate  transformer 
may  be  used  or  the  transformer  feature  embodied  in  the  motor, 
as  in  the  so-called  repulsion  type  of  motors.  This  gives  to  the 
alternating-current  commutator  motor  a  far  greater  variety  of 
connections  than  possessed  by  the  direct-current  motor. 

While  in  its  general  principle  of  operation  the  alternating- 
current  commutator  motor  is  identical  with  the  direct-current 
motor,  in  the  relative  proportioning  of  the  parts  a  great  differ- 
ence exists.  In  the  direct-current  motor,  voltage  is  consumed 
by  the  counter  e.m.f.  of  rotation,  which  represents  the  power 
output  of  the  motor,  and  by  the  resistance,  which  represents 
the  power  loss.  In  addition  thereto,  in  the  alternating-current 
motor  voltage  is  consumed  by  the  inductance,  which  is  wattless 
or  reactive  and  therefore  causes  a  lag  of  current  behind  the  vol- 
tage, that  is,  a  lowering  of  the  power-factor.  While  in  the  direct- 
current  motor  good  design  requires  the  combination  of  a  strong 
field  and  a  relatively  weak  armature,  so  as  to  reduce  the  armature 
reaction  on  the  field  to  a  minimum,  in  the  design  of  the  alter- 
nating-current motor  considerations  of  power-factor  predominate ; 
that  is,  to  secure  low  self -inductance  and  therewith  a  high  power- 
factor,  the  combination  of  a  strong  armature  and  a  weak  field  is 
required,  and  necessitates  the  use  of  methods  to  eliminate  the 
harmful  effects  of  high  armature  reaction. 

As  the  varying-speed  single-phase  commutator  motor  has 
found  an  extensive  use  as  railway  motor,  this  type  of  motor 
will  as  an  instance  be  treated  in  the  following,  and  the  other 
types  discussed  in  the  concluding  paragraphs. 

II.  Power-factor 

190.  In  the  commutating  machine  the  magnetic  field  flux  gen- 
erates the  e.m.f.  in  the  revolving  armature  conductors,  which 
gives  the  motor  output;  the  armature  reaction,  that  is,  the  mag- 
netic flux  produced  by  the  armature  current,  distorts  and  weakens 
the  field,  and  requires  a  shifting  of  the  brushes  to  avoid  sparking 
due  to  the  short-circuit  current  under  the  commutator  brushes, 
and  where  the  brushes  can  not  be  shifted,  as  in  a  reversible  motor, 
this  necessitates  the  use  of  a  strong  field  and  weak  armature  to 
keep  down  the  magnetic  flux  at  the  brushes.  In  the  alternating- 
current  motor  the  magnetic  field  flux  generates  in  the  armature 
conductors  by  their  rotation  the  e.m.f.  which  does  the  work  of 
the  motor,  but,  as  the  field  flux  is  alternating,  it  also  generates 


SINGLE-PHASE  COMMUTATOR  MOTORS         333 

• 
in  the  field  conductors  an  e.m.f.  of  self-inductance,  which  is  not 

useful  but  wattless,  and  therefore  harmful  in  lowering  the  power- 
factor,  hence  must  be  kept  as  low  as  possible. 

This  e.m.f.  of  self-inductance  of  the  field,  e0,  is  proportional 
to  the  field  strength,  <l>,  to  the  number  of  field  turns,  nQ)  and  to 
the  frequency,  /,  of  the  impressed  e.m.f. : 

eQ  =  27r/n0$  10~8,  (1) 

while  the  useful  e.m.f.  generated  by  the  field  in  the  armature 
conductors,  or  "  e.m.f.  of  rotation,"  e,  is  proportional  to  the  field 
strength,  <J>,  to  the  number  of  armature  turns,  HI,  and  to  the  fre- 
quency of  rotation  of  the  armature,  /0 : 

e  =  27r/0n1$10-8.  (2) 

This  later  e.m.f.,  e,  is  in  phase  with  the  magnetic  flux,  <f>,  and 
so  with  the  current,  i,  in  the  series  motor,  that  is,  is  a  power  e.m.f., 
while  the  e.m.f.  of  self-inductance,  eQ,  is  wattless,  or  in  quadrature 
with  the  current,  and  the  angle  of  lag  of  the  motor  current  thus 
is  given  by: 

tan  6  =      €.°  .  >  (3) 

e  +  ^r 

where  ir  =  voltage  consumed  by  the  motor  resistance.  Or  ap- 
proximately, since  ir  is  small  compared  with  e  (except  at  very 
low  speed) : 

tan  0  =  ~>  (4) 

e 

and,  substituting  herein  (1)  and  (2): 

tan  S  =  £  -°-  (5) 

Jo  n\ 

Small  angle  of  lag  and  therewith  good  power-factor  therefore 
require  high  values  of  /o  and  HI  and  low  values  of  /  and  n0. 

High  /o  requires  high  motor  speeds  and  as  large  number  of 
poles  as  possible.  Low  /  means  low  impressed  frequency;  there- 
fore 25  cycles  is  generally  the  highest  frequency  considered  for 
large  commutating  motors. 

High  HI  and  low  no  means  high  armature  reaction  and  low 
field  excitation,  that  is,  just  the  opposite  conditions  from  that 
required  for  good  commutator-motor  design. 

Assuming  synchronism,  /0  =  /,  as  average  motor  speed — 750 
revolutions  with  a  four-pole  25-cycle  motor — an  armature  reac- 


334 


ELECTRICAL  APPARATUS 


tion,  HI,  equal  to  the  field  excitation,  n0,  would  then  give  tan 
0  =  1,  0  =  45°,  or  70.7  per  cent,  power-factor;  that  is,  with  an 
armature  reaction  beyond  the  limits  of  good  motor  design,  the 

power-factor  is  still  too  low  for  use. 

The  armature,  however,  also  has  a 
self-inductance;  that  is,  the  magnetic 
flux  produced  by  the  armature  cur- 
rent as  shown  diagrammatically  in 
Fig.  155  generates  a  reactive  e.m.f.  in 
the  armature  conductors,  which  again 
lowers  the  power-factor.  While  this 
armature  self-inductance  is  low  with 
small  number  of  armature  turns,  it 
becomes  considerable  when  the  num- 
ber of  armature  turns,  n\,  is  large 
compared  with  the  field  turns,  n0. 

Let  (Ro  =  field  reluctance,  that 
is,  reluctance  of  the  magnetic 


FIG.  155. — Distribution  of 
main  field  and  field  of  arma- 
ture reaction. 


field  circuit,  and  (Ri 
(Ro 


-T-  =  the  armature  reluctance,  that  is, 


b  =  —  =  ratio  of  reluctances  of  the  armature  and  the  field  mag- 

(Hl 

netic  circuit;  then,  neglecting  magnetic  saturation,  the  field  flux 
is: 


(Ro 


the  armature  flux  is: 


nil 


(Ro 


and  the  e.m.f.  of  self -inductance  of  the  armature  circuit  is : 
61  = 


(6) 


(7) 


hence,  the  total  e.m.f.  of  self-inductance  of  the  motor,  or  wattless 
e.m.f.,  by  (1)  and  (7)  is: 


(8) 


SINGLE-PHASE  COMMUTATOR  MOTORS         335 

and  the  angle  of  lag,  6,  is  given  by  : 

tan  e  = 


e 

5  _L  7vn.2 

(9) 


/  n02  +  bni 


/o 
or,  denoting  the  ratio  of  armature  turns  to  field  turns  by: 


q  = 
tan  0 


=  f-  (l  +  bq)  ,  (10) 

JoW  / 


and  this  is  a  minimum;  that  is,  the  power-factor  a  maximum,  for: 

35  {tan  «}=(), 
or: 


- 


(ID 

Vb 
and  the  maximum  power-factor  of  the  motor  is  then  given  by: 

tan  9<>  =  /  -4='-  *>  (12) 

Jo  \/b 

Therefore  the  greater  b  is  the  higher  the  power-factor  that 
can  be  reached  by  proportioning  field  and  armature  so  that 


Since  6  is  the  ratio  of  armature  reluctance  to  field  reluctance, 
good  power-factor  thus  requires  as  high  an  armature  reluctance 
and  as  low  a  field  reluctance  as  possible;  that  is,  as  good  a  mag- 
netic field  circuit  and  poor  magnetic  armature  circuit  as  feasible. 
This  leads  to  the  use  of  the  smallest  air  gaps  between  field  and 
armature  which  are  mechanically  permissible.  With  an  air  gap 
of  0.10  to  0.15  in.  as  the  smallest  safe  value  in  railway  work,  b 
can  not  well  be  made  larger  than  about  4. 

Assuming,  then,  6  =  4,  gives  q  =  2,  that  is,  twice  as  many 
armattire.  turns  as  field  turns;  n\  =  2  n0. 

The  angle  of  lag  in  this  case  is,  by  (12),  at  synchronism  :  /0  =  /, 

tan  60  =  1, 

giving  a  power-factor  of  70.7  per  cent. 

It  follows  herefrom  that  it  is  not  possible,  with  a  mechanically 


336 


ELECTRICAL  APPARATUS 


safe  construction,  at  25  cycles  to  get  a  good  power-factor  at 
moderate  speed,  from  a  straight  series  motor,  even  if  such  a 
design  as  discussed  above  were  not  inoperative,  due  to  excessive 
distortion  and  therefore  destructive  sparking. 

Thus  it  becomes  necessary  in  the  single-phase  commutator 
motor  to  reduce  the  magnetic  flux  of  armature  reaction,  that  is, 
increase  the  effective  magnetic  reluctance  of  the  armature  far 
beyond  the  value  of  the  true  magnetic  reluctance.  This  is  ac- 
complished by  the  compensating  winding  devised  by  Eickemeyer, 
by  surrounding  the  armature  with  a  stationary  winding  closely 
adjacent  and  parallel  to  the  armature  winding,  and  energized  by 
a  current  in  opposite  direction  to  the  armature  current,  and  of 
the  same  m.m.f.,  that  is,  the  same  number  of  ampere-turns,  as 
the  armature  winding. 


FIG.  156. — Circuits  of  single- 
phase  commutator  motor. 


FIG.  157. — Massed  field  winding  and 
distributed  compensating  winding. 


191.  Every  single-phase  commutator  motor  thus  comprises  a 
field  winding,  F,  an  armature  winding,  A,  and  a  compensating 
winding,  C,  usually  located  in  the  pole  faces  of  the  field,  as  shown 
in  Figs.  156  and  157. 

The  compensating  winding,  C,  is  either  connected  in  series  (but 
in  reversed  direction)  with  the  armature  winding,  and  then  has 
the  same  number  of  effective  turns,  or  it  is  short-circuited  upon 
itself,  thus  acting  as  a  short-circuited  secondary  with  the  arma- 
ture winding  as  primary,  or  the  compensating  winding  is  ener- 
gized by  the  supply  current,  and  the  armature  short-circuited  as 


SINGLE-PHASE  COMMUTATOR  MOTORS         337 

secondary.  The  first  case  gives  the  conductively  compensated 
series  motor,  the  second  case  the  inductively  compensated  series 
motor,  the  third  case  the  repulsion  motor. 

In  the  first  case,  by  giving  the  compensating  winding  more 
turns  than  the  armature,  overcompensation,  by  giving  it  less 
turns,  undercompensation,  is  produced.  In  the  second  case 
always  complete  (or  practically  complete)  compensation  results, 
irrespective  of  the  number  of  turns  of  the  winding,  as  primary 
and  secondary  currents  of  a  transformer  always  are  opposite  in 
direction,  and  of  the  same  m.m.f.  (approximately),  and  in  the 
third  case  a  somewhat  less  complete  compensation. 

With  a  compensating  winding,  C,  of  equal  and  opposite  m.m.f. 
to  the  armature  winding,  A,  the  resultant  armature  reaction  is 
zero,  and  the  field  distortion,  therefore,  disappears;  that  is,  the 
ratio  of  the  armature  turns  to  field  turns  has  no  direct  effect  on 
the  commutation,  but  high  armature  turns  and  low  field  turns 
can  be  used.  The  armature  self -inductance  is  reduced  from  that 
corresponding  to  the  armature  magnetic  flux,  $1,  in  Fig.  155  to 
that  corresponding  to  the  magnetic  leakage  flux,  that  is,  the 
magnetic  flux  passing  between  armature  turns  and  compensating 
turns,  or  the  "slot  inductance,"  which  is  small,  especially  if  rela- 
tively shallow  armature  slots  and  compensating  slots  are  used. 

The  compensating  winding,  or  the  "cross  field,"  thus  fulfils 
the  twofold  purpose  of  reducing  the  armature  self -inductance  to 
that  of  the  leakage  flux,  and  of  neutralizing  the  armature  reac- 
tion and  thereby  permitting  the  use  of  very  high  armature 
ampere-turns. 

The  main  purpose  of  the  compensating  winding  thus  is  to  de- 
crease the  armature  self-inductance;  that  is,  increase  the  effect- 
ive armature  reluctance  and  thereby  its  ratio  to  the  field  reluc- 
tance, 6,  and  thus  permit  the  use  of  a  much  higher  ratio,  '(/  =  —, 

UQ 

before  maximum  power-factor  is  reached,  and  thereby  a  higher 
power-factor. 

Even  with  compensating  winding,  with  increasing  q,  ultimately 
a  point  is  reached  where  the  armature  self-inductance  equals 
the  field  self -inductance;  and  beyond  this  the  power-factor  again 
decreases.  It  becomes  possible,  however,  by  the  use  of  the  com- 
pensating winding,  to  reach,  with  a  mechanically  good  design, 
values  of  b  as  high  as  16  to  20. 

Assuming  b  =  16  gives,  substituted  in  (11)  and  (12): 

3  =  4; 

22 


338  ELECTRICAL  APPARATUS 

that  is,  four  times  as  many  armature  turns  as  field  turns,  n\  = 
4  n0  and : 

tan  0o  =  ~j\ 

4JO 

hence,  at  synchronism: 

/o  =  /  :  tan  00  =  0.5,  or  89  per  cent,  power-factor. 

At  double  synchronism,  which  about  represents  maximum  motor 
speed  at  25  cycles : 

/o  =  2/  :  tan  00  =  0.25,  or  98  per  cent,  power-factor; 

that  is,  very  good  power-factors  can  be  reached  in  the  single- 
phase  commutator  motor  by  the  use  of  a  compensating  winding, 
far  higher  than  are  possible  with  the  same  air  gap  in  polyphase 
induction  motors. 

III.  Field  Winding  and  Compensating  Winding 

192.  The  purpose  of  the  field  winding  is  to  produce  the  maxi- 
mum magnetic  flux,  <£,  with  the  minimum  number  of  turns,  n0. 
This  requires  as  large  a  magnetic  section,  especially  at  the  air 
gap,  as  possible.  Hence,  a  massed  field  winding  with  definite 
polar  projections  of  as  great  pole  arc  as  feasible,  as  shown  in  Fig. 
157,  gives  a  better  power-factor  than  a  distributed  field  winding. 

The  compensating  winding  must  be  as  closely  adjacent  to  the 
armature  winding  as  possible,  so  as  to  give  minimum  leakage 
flux  between  armature  conductors  and  compensating  conductors, 
and  therefore  is  a  distributed  winding,  located  in  the  field  pole 
faces,  as  shown  in  Fig.  157. 

The  armature  winding  is  distributed  over  the  whole  circum- 
ference of  the  armature,  but  the  compensating  winding  only  in 
the  field  pole  faces.  With  the  same  ampere-turns  in  armature 
and  compensating  winding,  their  resultant  ampere-turns  are 
equal  and  opposite,  and  therefore  neutralize,  but  locally  the  two 
windings  do  not  neutralize,  due  to  the,  difference  in  the  distribu- 
tion curves  of  their  m.m.fs.  The  m.m.f.  of  the  field  winding  is 
constant  over  the  pole  faces,  and  from  one  pole  corner  to  the  next 
pole  corner  reverses  in  direction,  as  shown  diagrammatically 
by  F  in  Fig.  158,  which  is  the  development  of  Fig.  157.  The 
m.m.f.  of  the  armature  is  a  maximum  at  the  brushes,  midway 
between  the  field  poles,  as  shown  by  A  in  Fig.  158,  and  from  there 
decreases  to  zero  in  the  center  of  the  field  pole.  The  m.m.f.  of 


SINGLE-PHASE  COMMUTATOR  MOTORS         339 

the  compensating  winding,  however,  is  constant  in  the  space 
from  pole  corner  to  pole  corner,  as  shown  by  C  in  Fig.  158,  and 
since  the  total  m.m.f.  of  the  compensating  winding  equals  that 
of  the  armature,  the  armature  m.m.f.  is  higher  at  the  brushes, 
the  compensating  m.m.f.  higher  in  front  of  the  field  poles,  as 
shown  by  curve  R  in  Fig.  158,  which  is  the  difference  between 
A  and  C;  that  is,  with  complete  compensation  of  the  resultant 
armature  and  compensating  winding,  locally  undercompensation 
exists  at  the  brushes,  overcompensation  in  front  of  the  field 


FIG.  158.~Distribution  of  m.m.f.  in  compensated  motor. 

poles.  The  local  undercompensated  armature  reaction  at  the 
brushes  generates  an  e.m.f.  in  the  coil  short-circuited  under  the 
brush,  and  therewith  a  short-circuit  current  of  commutation 
and  sparking.  In  the  conductively  compensated  motor,  this  can 
be  avoided  by  overcompensation,  that  is,  raising  the  flat  top  of 
the  compensating  m.m.f.  to  the  maximum  armature  m.m.f.,  but 
this  results  in  a  lowering  of  the  power-factor,  due  to  the  self- 
inductive  flux  of  overcompensation,  and  therefore  is  undesirable. 
193.  To  get  complete  compensation  even  locally  requires  the 
compensating  winding  to  give  the  same  distribution  curve  as  the 
armature  winding,  or  inversely.  The  former  is  accomplished  by 
distributing  the  compensating  winding  around  the  entire  cir- 
cumference of  the  armature,  as  shown  in  Fig.  159.  This,  how- 
ever, results  in  bringing  the  field  coils  further  away  from  the 
armature  surface,  and  so  increases  the  magnetic  stray  flux  of  the 
field  winding,  that  is,  the  magnetic  flux,  which  passes  through 
the  field  coils,  and  there  produces  a  reactive  voltage  of  self-in- 


340 


ELECTRICAL  APPARATUS 


ductance,  but  does  not  pass  through  the  armature  conductors, 
and  so  does  no  work;  that  is,  it  lowers  the  power  factor,  just  as 
over  compensation  would  do.  The  distribution  curve  of  the 

armature   winding  can,  however,  be 

f~  ~\        made  equal  to  that  of  the  compen- 

sating winding,  and  therewith  local 
complete  compensation  secured,  by 
using  a  fractional  pitch  armature 
winding  of  a  pitch  equal  to  the  pole 
arc.  In  this  case,  in  the  space  be- 
tween the  pole  corners,  the  currents 
are  in  opposite  direction  in  the 
upper  and  the  lower  layer  of  con- 
ductors in  each  armature  slot,  as 
shown  in  Fig.  160,  and  thus  neutralize 
magnetically;  that  is,  the  armature 
reaction  extends  only  over  the  space 
of  the  armature  circumference  covered 
by  the  pole  arc,  where  it  is  neutralized 
by  the  compensating  winding  in  the  pole  face. 

To  produce  complete  compensation  even  locally,  without  im- 
pairing the  power-factor,  therefore,  requires  a  fractional-pitch 


FIG.  159. — Completely 
distributed  compensating 
winding. 


FIG.  160. — Fractional  pitch  arma- 
ture winding. 


FIG.    161. — Repulsion    motor    with 
massed  winding. 


armature  winding,  of  a  pitch  equal  to  the  field  pole  arc,  or  some 
equivalent  arrangement. 

Historically,  the  first  compensated  single-phase  commutator 
motors,  built  about  20  years  ago,  were  Prof.  Elihu  Thomson's 
repulsion  motors.  In  these  the  field  winding  and  compensating 


SINGLE-PHASE  COMMUTATOR  MOTORS         341 


winding  were  massed  together  in  a  single  coil,  as  shown  diagram- 
matically  in  Fig.  161.  Repulsion  motors  are  still  occasionally 
.built  in  which  field  and  compensating  coils  are  combined  in  a 
single  distributed  winding,  as  shown  in  Fig.  162.  Soon  after  the 
first  repulsion  motor,  conductively  and  inductively  compensated 
series  motors  were  built  by  Eickemeyer,  with  a  massed  field 
winding  and  a  separate  compensating  winding,  or  cross  coil, 
either  as  single  coil  or  turn  or  distributed  in  a  number  of  coils  or 
turns,  as  shown  diagrammatically  in  Fig.  163,  and  by  W.  Stanley. 


t: ::     '^m  C 


FIG.  162. — Repulsion  motor  with 
distributed  winding. 


FIG.  163. — Eickemeyer  inductively 
compensated  series  motor. 


For  reversible  motors,  separate  field  coils  and  compensating 
coils  are  always  used,  the  former  as  massed,  the  latter  as  dis- 
tributed winding,  since  in  reversing  the  direction  of  rotation 
either  the  field  winding  alone  must  be  reversed  or  armature  and 
compensating  winding  are  reversed  while  the  field  winding  re- 
mains unchanged. 

IV.  Types  of  Varying-speed  Single-phase  Commutator  Motors 

194.  The  armature  and  compensating  windings  are  in  induc- 
tive relations  to  each  other.  In  the  single-phase  commutator 
motor  with  series  characteristic,  armature  and  compensating 
windings  therefore  can  be  connected  in  series  with  each  other,  or 
the  supply  voltage  impressed  upon  the  one,  the  other  closed  upon 
itself  as  secondary  circuit,  or  a  part  of  the  supply  voltage  im- 
pressed upon  the  one,  and  another  part  upon  the  other  circuit, 
and  in  either  of  these  cases  the  field  winding  may  be  connected 
in  series  either  to  the  compensating  winding  or  to  the  armature 
winding.  This  gives  the  motor  types,  denoting  the  armature  by 


342  ELECTRICAL  APPARATUS 


(D 


(4) 


(2) 


(5) 


(3) 


(6) 


(7) 


FIG.  164. — Types  of  alternating-current  commutating  motors. 


SINGLE-PHASE  COMMUTATOR  MOTORS 


343 


A}  the  compensating  winding  by  C,  and  the  field  winding  by  F, 
shown  in  Fig.  164. 


Primary 

A  +F 

A  +  C  +  F 

A+F 
A 


C  +  F 
C 

A+F,C 
A,  C  +  F 


Secondary 

C 

C  +  F 


A 

A  +  F 


Series  motor. 

Conductively  compensated 
series  motor.  (1) 

Inductively  compen  sated 
series  motor.  (2) 

Inductively  compensated 
series  motor  with  second- 
ary excitation,  or  inverted 
repulsion  motor.  (3) 

Repulsion  motor.     (4) 

Repulsion  motor  with  sec- 
ondary excitation.  (5) 

Series  repulsion  motors. 
(6)  (7) 


Since  in  all  these  motor  types  all  three  circuits  are  connected 
directly  or  inductively  in  series  with  each  other,  they  all  have 
the  same  general  characteristics  as  the  direct-current  series 
motor;  that  is,  a  speed  which  increases  with  a  decrease  of  load, 
and  a  torque  per  ampere  input  which  increases  with  increase  of 
current,  and  therefore  with  decrease  of  speed,  and  the  different 
motor  types  differ  from  each  other  only  by  their  commutation 
as  affected  by  the  presence  or  absence  of  a  magnetic  flux  at  the 
brushes,  and  indirectly  thereby  in  their  efficiency  as  affected  by 
commutation  losses. 

In  the  conductively  compensated  series  motor,  by  the  choice 
of  the  ratio  of  armature  and  compensating  turns,  overcompensa- 
tion,  complete  compensation,  or  undercompensation  can  be  pro- 
duced. In  all  the  other  types,  armature  and  compensating 
windings  are  in  inductive  relation,  and  the  compensation  there- 
fore approximately  complete. 

A  second  series  of  motors  of  the  same  varying  speed  charac- 
teristics results  by  replacing  the  stationary  field  coils  by  arma- 
ture excitation,  that  is,  introducing  the  current,  either  directly 
or  by  transformer,  into  the  armature  by  means  of  a  second  set 
of  brushes  at  right  angles  to  the  main  brushes.  Such  motors 
are  used  to  some  extent  abroad.  They  have  the  disadvantage  of 


344 


ELECTRICAL  APPARATUS 


FIG.  165. — Type  of  alternating-cur- 
rent commutating  motor. 


requiring  two  sets  of  brushes,  but  the  advantage  that  their 
power-factor  can  be  controlled  and  above  synchronism  even 
leading  current  produced.  Fig.  165  shows  diagrammatically  such 
a  motor,  as  designed  by  Winter-Eichberg-Latour,  the  so-called 
compensated  repulsion  motor.  In  this  case  compensated  means 
compensated  for  power-factor. 

The  voltage  which  can  be  used  in  the  motor  armature  is  limited 
by  the  commutator:  the  voltage  per  commutator  segment  is 
limited  by  the  problem  of  sparkless  commutation,  the  number 

of  commutator  segments  from 
brush  to  brush  is  limited  by 
mechanical  consideration  of 
commutator  speed  and  width 
of  segments.  In  those  motor 
types  in  which  the  supply  cur- 
rent traverses  the  armature,  the 
supply  voltage  is  thus  limited 
to  values  even  lower  than  in 
the  direct-current  motor,  while 
in  the  repulsion  motor  (4  and 
5),  in  which  the  armature  is  the 
secondary  circuit,  the  armature  voltage  is  independent  of  the 
supply  voltage,  so  can  be  chosen  to  suit  the  requirements  of 
commutation,  while  the  motor  can  be  built  for  any  supply 
voltage  for  which  the  stator  can  economically  be  insulated. 

Alternating-current  motors  as  well  as  direct-current  series 
motors  can  be  controlled  by  series  parallel  connection  of  two  or 
more  motors.  Further  control,  as  in  starting,  with  direct-current 
motors  is  carried  out  by  rheostat,  while  with  alternating-current 
motors  potential  control,  that  is,  a  change  of  supply  voltage  by 
transformer  or  autotransformer,  offers  a  more  efficient  method 
of  control.  By  changing  from  one  motor  type  to  another  motor 
type,  potential  control  can  be  used  in  alternating-current  motors 
without  any  change  of  supply  voltage,  by  appropriately  choosing 
the  ratio  of  turns  of  primary  and  secondary  circuit.  For  in- 
stance, with  an  armature  wound  for  half  the  voltage  and  thus 

twice  the  current  as  the  compensating  winding  (ratio  of  turns 

-  »  2  V,  a  change  of  connection  from  type  3  to  type  2,  or  from 
type  5  to  type  4,  results  in  doubling  the  field  current  and  there- 


SINGLE-PHASE  COMMUTATOR  MOTORS         345 

with  the  field  strength.  A  change  of  distribution  of  voltage  be- 
tween the  two  circuits,  in  types  6  and  7,  with  A  and  C  wound 
for  different  voltages,  gives  the  same  effect  as  a  change  of  supply 
voltage,  and  therefore  is  used  for  motor  control. 

195.  In  those  motor  types  in  which  a  transformation  of  power 
occurs  between  compensating  winding,  C,  and  armature  winding, 
A,  a  transformer  flux  exists  in  the  direction  of  the  brushes,  that 
is,  at  right  angles  to  the  field  flux.  In  general,  therefore,  the 
single-phase  commutator  motor  contains  two  magnetic  fluxes  in 
quadrature  position  with  each  other,  the  main  flux  or  field  flux, 
<J>,  in  the  direction  of  the  axis  of  the  field  coils,  or  at  right  angles 
to  the  armature  brushes,  and  the  quadrature  flux,  or  transformer 
flux,  or  commutating  flux,  $1,  in  line  with  the  armature  brushes, 
or  in  the  direction  of  the  axis  of  the  compensating  winding,  that 
is,  at  right  angles  (electrical)  with  the  field  flux. 

The  field  flux,  3>,  depends  upon  and  is  in  phase  with  the  field 
current,  except  as  far  as  it  is  modified  by  the  magnetic  action  of 
the  short-circuit  current  in  the  armature  coil  under  the  commu- 
tator brushes. 

In  the  conductively  compensated  series  motor,  1,  the  quad- 
rature flux  is  zero  at  complete  compensation,  and  in  the  direc- 
tion of  the  armature  reaction  with  undercompensation,  in  oppo- 
sition to  the  armature  reaction  at  overcompensation,  but  in 
either  case  in  phase  with  the  current  and  so  approximately  with 
the  field. 

In  the  other  motor  types,  whatever  quadrature  flux  exists  is 
not  in  phase  with  the  main  flux,  but  as  transformer  flux  is  due 
to  the  resultant  m.m.f.  of  primary  and  secondary  circuit. 

In  a  transformer  with  non-inductive  or  nearly  non-inductive 
secondary  circuit,  the  magnetic  flux  is  nearly  90°  in  time  phase 
behind  the  primary  current,  a  little  over  90°  ahead  of  the  sec- 
ondary current,  as  shown  in  transformer  diagram,  Fig.  166. 

In  a  transformer  with  inductive  secondary,  the  magnetic  flux 
is  less  than  90°  behind  the  primary  current,  more  than  90°  ahead 
of  the  secondary  current,  the  more  so  the  higher  is  the  inductivity 
of  the  secondary  circuit,  as  shown  by  the  transformer  diagram, 
Fig.  166. 

Herefrom  it  follows  that : 

In  the  inductively  compensated  series  motor,  2,  the  quad- 
rature flux  is  very  small  and  practically  negligible,  as  very  little 
voltage  is  consumed  in  the  low  impedance  of  the  secondary  cir- 
cuit, C;  whatever  flux-there  is,  lags  behind  the  main  flux. 


346 


ELECTRICAL  APPARATUS 


In  the  inductively  compensated  series  motor  with  secondary 
excitation,  or  inverted  repulsion  motor,  3,  the  quadrature  flux, 
$1,  is  quite  large,  as  a  considerable  voltage  is  required  for  the 
field  excitation,  especially  at  moderate  speeds  and  therefore  high 
currents,  and  this  flux,  3>i,  lags  behind  the  field  flux,  <l>,  but  this 
lag  is  very  much  less  than  90°,  since  the  secondary  circuit  is 


*1    l  ~3> 

FIG.  166. — Transformer  diagram,  inductive  and  non-inductive  load. 

highly  inductive;  the  motor  field  thus  corresponding  to  the  con- 
ditions of  the  transformer  diagram,  Fig.  166.  As  result  hereof, 
the  commutation  of  this  type  of  motor  is  very  good,  flux,  3>i, 
having  the  proper  phase  and  intensity  required  for  a  commu- 
tating  flux,  as  will  be  seen  later,  but  the  power-factor  is  poor. 

In  the  repulsion  motor,  4,  the  quadrature  flux  is  very  consid- 
erable, since  all  the  voltage  consumed  by  the  rotation  of  the 
armature  is  induced  in  it  by  transformation  from  the  compen- 


SINGLE-PHASE  COMMUTATOR  MOTORS         347 

sating  winding,  and  this  quadrature  flux,  $1,  lags  nearly  90°  be- 
hind the  main  flux,  $,  since  the  secondary  circuit  is  nearly  non- 
inductive,  especially  at  speed. 

In  the  repulsion  motor  with  secondary  excitation,  5,  the  quad- 
rature flux,  $1,  is  also  very  large,  and  practically  constant,  corre- 
sponding to  the  impressed  e.m.f.,  but  lags  considerably  less  than 
90°  behind  the  main  flux,  $,  the  secondary  circuit  being  induct- 
ive, since  it  contains  the  field  coil,  F.  The  lag  of  the  flux,  <i>i, 
increases  with  increasing  speed,  since  with  increasing  speed  the 
e.m.f.  of  rotation  of  the  armature  increases,  the  e.m.f.  of  self- 
inductance  of  the  field  decreases,  due  to  the  decrease  of  current, 
and  the  circuit  thus  becomes  less  inductive. 

The  series  repulsion  motors  6  and  7,  give  the  same  phase  rela- 
tion of  the  quadrature  flux,  $1,  as  the  repulsion  motors,  5  and  6, 
but  the  intensity  of  the  quadrature  flux,  3>i,  is  the  less  the  smaller 
the  part  of  the  supply  voltage  which  is  impressed  upon  the  com^ 
pensating  winding. 

V.  Commutation 

196.  In  the  commutator  motor,  the  current  in  each  armature 
coil  or  turn  reverses  during  its  passage  under  the  brush.  In  the 
armature  coil,  while  short-circuited  by  the  commutator  brush, 
the  current  must  die  out  to  zero  and  then  increase  again  to  its 
original  value  in  opposite  direction.  The  resistance  of  the  arma- 
ture coil  and  brush  contact  accelerates,  the  self-inductance  re- 
tards the  dying  out  of  the  current,  and  the  former  thus  assists, 
the  latter  impairs  commutation.  If  an  e.m.f.  is  generated  in 
the  armature  coil  by  its  rotation  while  short-circuited  by  the 
commutator  brush,  this  e.m.f.  opposes  commutation,  that  is, 
retards  the  dying  out  of  the  current,  if  due  to  the  magnetic  flux 
of  armature  reaction,  and  assists  commutation  by  reversing  the 
armature  current,  if  due  to  the  magnetic  flux  of  overcompensa- 
tion,  that  is,  a  magnetic  flux  in  opposition  to  the  armature 
reaction.  • 

Therefore,  in  the  direct-current  commutator  motor  with  high 
field  strength  and  low  armature  reaction,  that  is,  of  negligible 
magnetic  flux  of  armature  reaction,  fair  commutation  is  produced 
with  the  brushes  set  midway  between  the  field  poles — that  is, 
in  the  position  where  the  armature  coil  which  is  being  commu- 
tated  encloses  the  full  field  flux  and  therefore  cuts  no  flux  and 
has  no  generated  e.m.f. — by  using  high-resistance  carbon  brushes, 


348  ELECTRICAL  APPARATUS 

as  the  resistance  of  the  brush  contact,  increasing  when  the  arma- 
ture coil  begins  to  leave  the  brush,  tends  to  reverse  the  current. 
Such  " resistance  commutation"  obviously  can  not  be  perfect; 
perfect  commutation,  however,  is  produced  by  impressing  upon 
the  motor  armature  at  right  angles  to  the  main  field,  that  is,  in 
the  position  of  the  commutator  brushes,  a  magnetic  field  oppo- 
site to  that  of  the  armature  reaction  and  proportional  to  the 
armature  current.  Such  a  field  is  produced  by  overcompensa- 
tion  or  by  the  use  of  a  commutating  pole  or  interpole. 

As  seen  in  the  foregoing,  in  the  direct-current  motor  the  counter 
e.m.f.  of  self-inductance  of  commutation  opposes  the  reversal  of 
current  in  the  armature  coil  under  the  commutator  brush,  and 
this  can  be  mitigated  in  its  effect  by  the  use  of  high-resistance 
brushes,  and  overcome  by  the  commutating  field  of  overcompen- 
sation.  In  addition  hereto,  however,  in  the  alternating-current 
commutator  motor  an  e.m.f.  is  generated  in  the  coil  short-cir- 
cuited under  the  brush,  by  the  alternation  of  the  magnetic  flux, 
and  this  e.m.f.,  which  does  not  exist  in  the  direct-current  motor, 
makes  the  problem  of  commutation  of  the  alternating-current 
motor  far  more  difficult.  In  the  position  of  commutation  no 
e.m.f.  is  generated  in  the  armature  coil  by  its  rotation  through 
the  magnetic  field,  as  in  this  position  the  coil  encloses  the  maxi- 
mum field  flux;  but  as  this  magnetic  flux  is  alternating,  in  this 
position  the  e.m.f.  generated  by  the  alternation  of  the  flux  en- 
closed by  the  coil  is  a  maximum.  This  " e.m.f.  of  alternation" 
lags  in  time  90°  behind  the  magnetic  flux  which  generates  it,  is 
proportional  to  the  magnetic  flux  and  to  the  frequency,  but  is 
independent  of  the  speed,  hence  exists  also  at  standstill,  while 
the  "e.m.f.  of  rotation " — which  is  a  maximum  in  the  position 
of  the  armature  coil  midway  between  the  brushes,  or  parallel  to 
the  field  flux — is  in  phase  with  the  field  flux  and  proportional 
thereto  and  to  the  speed,  but  independent  of  the  frequency.  In 
the  alternating-current  commutator  motor,, no  position  therefore 
exists  in  which  the  armature  coil  is  free  from  a  generated  e.m.f., 
but  in  the  position  parallel  to  the  field,  or  midway  between  the 
brushes,  the  e.m.f.  of  rotation,  in  phase  with  the  field  flux,  is  a 
maximum,  while  the  e.m.f.  of  alternation  is  zero,  and  in  the  posi- 
tion under  the  commutator  brush,  or  enclosing  the  total  field 
flux,  the  e.m.f.  of  alternation,  in  electrical  space  quadrature  with 
the  field  flux,  is  a  maximum,  the  e.m.f.  of  rotation  absent,  while 
in  any  other  position  of  the  armature  coil  its  generated  e.m.f.  has 


SINGLE-PHASE  COMMUTATOR  MOTORS         349 

a  component  due  to  the  rotation — a  power  e.m.f. — and  a  com- 
ponent due  to  the  alternation — a  reactive  e.m.f.     The  armature 
coils  of  an  alternating-current  commutator  motor,  therefore,  are 
the  seat  of  a  system  of  polyphase  e.m.fs.,  and  at  synchronism 
the  polyphase  e.m.fs.  generated  in  all  armature  coils  are  equal, 
above   synchronism   the    e.m.f.    of   rotation    is  greater,   while 
below  synchronism  the  e.m.f.  of  alternation  is  greater,  and  in 
the  latter  case  the  brushes  thus  stand  at  that  point  of  the  com- 
mutator where  the  voltage  between  commutator  segments  is  a 
maximum.     This  e.m.f.  of  alternation,  short-circuited  by  the 
armature  coil  in  the  position  of  commutation,  if  not  controlled, 
causes  a  short-circuit  current  of  excessive  value,  and  therewith 
destructive  sparking;  hence,  in  the  alternating-current  commuta- 
tor motor  it  is  necessary  to  provide  means  to  control  the  short- 
circuit  current  under  the  commutator  brushes,  which  results  from 
the  alternating  character  of  the  magnetic  flux,  and  which  does 
not  exist  in  the  direct-current  motor;  that  is,  in  the  alternating- 
current  motor  the  armature  coil  under  the  brush  is  in  the  posi- 
tion of  a  short-circuited  secondary,  with  the  field  coil  as  primary 
of  a  transformer;  and  as  in  a  transformer  primary  and  secondary 
ampere-turns  are  approximately  equal,  if  n0  =  number  of  field 
turns  per  pole  and  i  =  field  current,  the  current  in  a  single  arma- 
ture turn,  when  short-circuited  by  the  commutator  brush,  tends 
to  become  i0  =  n0i,  that  is,  many  times  full-load  current;  and 
as  this  current  is  in  opposition,  approximately,  to  the  field  cur- 
rent, it  would  demagnetize  the  field;  that  is,  the  motor  field 
vanishes,  or  drops  far  down,  and  the  motor  thus  loses  its  torque. 
Especially  is  this  the  case  at  the  moment  of  starting;  at  speed, 
the  short-circuit  current  is  somewhat  reduced  by  the  self-induc- 
tance of  the  armature  turn.     That  is,  during  the  short  time 
during  which  the  armature  turn  or  coil  is  short-circuited  by  the 
brush  the  short-circuit  current  can  not  rise  to  its  full  value,  if 
the  speed  is  considerable,  but  it  is  still  sufficient  to  cause  destruc- 
tive sparking. 

197.  The  character  of  the  commutation  of  the  motor,  and 
therefore  its  operativeness,  thus  essentially  depends  upon  the 
value  and  the  phase  of  the  short-circuit  currents  under  the  com- 
mutator brushes.  An  excessive  short-circuit  current  gives  de- 
structive sparking  by  high-current  density  under  the  brushes 
and  arcing  at  the  edge  of  the  brushes  due  to  the  great  and  sud- 
den change  of  current  in  the  armature  coil  when  leaving  the 


350 


ELECTRICAL  APPARATUS 


brush.  But  even  with  a  moderate  short-circuit  current,  the 
sparking  at  the  commutator  may  be  destructive  and  the  motor 
therefore  inoperative,  if  the  phase  of  the  short-circuit  current 
greatly  differs  from  that  of  the  current  in  the  armature  coil  after 
it  leaves  the  brush,  and  so  a  considerable  and  sudden  change  of 


^g"* 

—  •  — 

• 

—  —  - 

•            — 

•             • 

V( 

LTS 
0.25 

0  ?0 

X 

•^ 

/ 

0,15 

/ 

0.7 

/ 

0  05 

2 

0      4 

0      6 

0      8 

0     100     1 

AMP1.  PER  SQ. 
!0    140    160     1 

N. 

BO     2 

)0     2 

>0     2 

10    2 

30    2 

50     3 

JO 

FIG.  167. — E.m.f.  consumed  at  contact  of  copper  brush. 

current  must  take  place  at  the  moment  when  the  armature  coil 
leaves  the  brush.  That  is,  perfect  commutation  occurs,  if  the 
short-circuit  current  in  the  armature  coil  under  the  commutator 
brush  at  the  moment  when  the  coil  leaves  the  brush  has  the 
same  value  and  the  same  phase  as  the  main-armature  current  in 


sa—  —  = 



_      — 

V( 

__        — 

3LTS 
_L6. 

1  4 

^-  —  ' 

.—  -  — 

..  —  —  - 

x 

^ 

1  2 

/ 

x^ 

i  n 

/ 

0  8 

/ 

f 

0  6 

/ 

/ 

0  4 

/ 

/ 

0  ?. 

/\ 

0      5 

0      3 

0      4 

0       E 

0      6 

AMP.  PER  SQ. 
0      70      80      < 

IN. 
0      1( 

)0     1 

LO     1 

20     1 

JO     1 

:0         1 

>0 

FIG.  168. — E.m.f.  consumed  at  contact  of  high-resistance  carbon  brush. 

the  coil  after  leaving  the  brush.  The  commutation  of  such  a 
motor  therefore  is  essentially  characterized  by  the  difference 
between  the  main-armature  current  after,  and  the  short-circuit 
current  before  leaving  the  brush.  The  investigation  of  the  short- 
circuit  current  under  the  commutator  brushes  therefore  is  of 


SINGLE-PHASE  COMMUTATOR  MOTORS      .  351 

fundamental  importance  in  the  study  of  the  alternating-current 
commutator  motor,  and  the  control  of  this  short-circuit  current 
the  main  problem  of  alternating-current  commutator  motor 
design. 

Various  means  have  been  proposed  and  tried  to  mitigate  or 
eliminate  the  harmful  effect  of  this  short-circuit  current,  as  high 
resistance  or  high  reactance  introduced  into  the  armature  coil 
during  commutation,  or  an  opposing  e.m.f.  either  from  the  out- 
side, or  by  a  commutating  field. 

High-resistance  brush  contact,  produced  by  the  use  of  very 
narrow  carbon  brushes  of  high  resistivity,  while  greatly  improv- 
ing the  commutation  and  limiting  the  short-circuit  current  so 
that  it  does  not  seriously  demagnetize  the  field  and  thus  cause 
the  motor  to  lose  its  torque,  is  not  sufficient,  for  the  reason  that 
the  resistance  of  the  brush  contact  is  not  high  enough  and  also  is 
not  constant.  The  brush  contact  resistance  is  not  of  the  nature 
of  an  ohmic  resistance,  but  more  of  the  nature  of  a  counter 
e.m.f.;  that  is,  for  large  currents  the  potential  drop  at  the  brushes 
becomes  approximately  constant,  as  seen  from  the  volt-ampere 
characteristics  of  different  brushes  given  in  Figs.  167  and  168. 
Fig.  167  gives  the  voltage  consumed  by  the  brush  contact  of  a 
copper  brush,  with  the  current  density  as  abscissae,  while  Fig. 
168  gives  the  voltage  consumed  by  a  high-resistance  carbon 
brush,  with  the  current  density  in  the  brush  as  abscissae.  It  is 
seen  that  such  a  resistance,  which  decreases  approximately  in- 
versely proportional  to  the  increase  of  current,  fails  in  limiting 
the  current  just  at  the  moment  where  it  is  most  required,  that 
s,  at  high  currents. 

Commutator  Leads 

198.  Good  results  have  been  reached  by  the  use  of  metallic 
resistances  in  the  leads  between  the  armature  and  the  commuta- 
tor. As  shown  diagrammatically  in  Fig.  169,  each  commutator 
segment  connects  to  the  armature,  A ,  by  a  high  non-inductive 
resistance,  CB,  and  thus  two  such  resistances  are  always  in  the 
circuit  of  the  armature  coil  short-circuited  under  the  brush,  but 
also  one  or  two  in  series  with  the  armature  main  circuit,  from 
brush  to  brush.  While  considerable  power  may  therefore  be 
consumed  in  these  high-resistance  leads,  nevertheless  the  effi- 
ciency of  the  motor  is  greatly  increased  by  their  use ;  that  is,  the 
reduction  in  the  loss  of  power  at  the  commutator  by  the  reduction 


352 


ELECTRICAL  APPARATUS 


of  the  short-circuit  current  usually  is  far  greater  than  the  waste 
of  power  in  the  resistance  leads.  To  have  any  appreciable  effect, 
the  resistance  of  the  commutator  lead  must  be  far  higher  than 
that  of  the  armature  coil  to  which  it  connects.  Of  the  e.m.f. 
of  rotation,  that  is,  the  useful  generated  e.m.f.,  the  armature  re- 
sistance consumes  only  a  very  small  part,  a  few  per  cent.  only. 
The  e.m.f.  of  alternation  is  of  the  same  magnitude  as  the  e.m.f. 
of  rotation — higher  below,  lower  above  synchronism.  With  a 
short-circuit  current  equal  to  full-load  current,  the  resistance  of 


FIG.  169. — Commutation  with  resistance  leads. 

the  short-circuit  coil  would  consume  only  a  small  part  of  the 
e.m.f.  of  alternation,  and  to  consume  the  total  e.m.f.  the  short- 
circuit  current  therefore  would  have  to  be  about  as  many  times 
larger  than  the  normal  armature  current  as  the  useful  generated 
e.m.f.  of  the  motor  is  larger  than  the  resistance  drop  in  the  arma- 
ture. Long  before  this  value  of  short-circuit  current  is  reached 
the  magnetic  field  would  have  disappeared  by  the  demagnetizing 
force  of  the  short-circuit  current,  that  is,  the  motor  would  have 
lost  its  torque. 

To  limit  the  short-circuit  current  under  the  brush  to  a  value 
not  very  greatly  exceeding  full-load  current,  thus  requires  a  re- 
sistance of  the  lead,  many  times  greater  than  that  of  the  armature 
coil.  The  i2r  in  the  lead,  and  thus  the  heat  produced  in  it,  then, 
is  many  times  greater  than  that  in  the  armature  coil.  The  space 
available  for  the  resistance  lead  is,  however,  less  than  that  avail- 
able for  the  armature  coil. 

It  is  obvious  herefrom  that  it  is  not  feasible  to  build  these 
resistance  leads  so  that  each  lead  can  dissipate  continuously,  or 
even  for  any  appreciable  time,  without  rapid  self-destruction, 
the  heat  produced  in  it  while  in  circuit. 

When  the  motor  is  revolving,  even  very  slowly,  this  is  not  nec- 
essary, since  each  resistance  lead  is  only  a  very  short  time  in 


SINGLE-PHASE  COMMUTATOR  MOTORS        353 

circuit,  during  the  moment  when  the  armature  coils  connecting 

to  it  are  short-circuited  by  the  brushes ;  that  is,  if  HI  =  number  of 

2 
armature  turns  from  brush  to  brush,  the  lead  is  only  —  of  the 

77-1 

time  in  circuit,  and  though  excessive  current  densities  in  mate- 
rials of  high  resistivity  are  used,  the  heating  is  moderate.  In 
starting  the  motor,  however,  if  it  does  not  start  instantly,  the 
current  continues  to  flow  through  the  same  resistance  leads,  and 
thus  they  are  overheated  and  destroyed  if  the  motor  does  not 
start  promptly.  Hence  care  has  to  be  taken  not  to  have  such 
motors  stalled  for  any  appreciable  time  with  voltage  on. 

The  most  serious  objection  to  the  use  of  high-resistance  leads, 
therefore,  is  their  liability  to  self-destruction  by  heating  if  the 
motor  fails  to  start  immediately,  as  for  instance  in  a  railway 
motor  when  putting  the  voltage  on  the  motor  before  the  brakes 
are  released,  as  is  done  when  starting  on  a  steep  up-grade  to 
keep  the  train  from  starting  to  run  back. 

Thus  the  advantages  of  resistance  commutator  leads  are  the 
improvement  in  commutation  resulting  from  the  reduced  short- 
circuit  current,  and  the  absence  o  fa  serious  demagnetizing  effect 
on  the  field  at  the  moment  of  starting,  which  would  result  from 
an  excessive  short-circuit  current  under  the  brush,  and  such 
leads  are  therefore  extensively  used ;  their  disadvantage,  however, 
is  that  when  they  are  used  the  motor  must  be  sure  to  start  im- 
mediately by  the  application  of  voltage,  otherwise  they  are  liable 
to  be  destroyed. 

It  is  obvious  that  even  with  high-resistance  commutator  leads 
the  commutation  of  the  motor  can  not  be  as  good  as  that  of  the 
motor  on  direct-current  supply;  that  is,  such  an  alternating- 
current  motor  inherently  is  more  or  less  inferior  in  commutation 
to  the  direct-current  motor,  and  to  compensate  for  this  effect 
far  more  favorable  constants  must  be  chosen  in  the  motor  design 
than  permissible  with  a  direct-current  motor,  that  is,  a  lower 
voltage  per  commutator  segment  and  lower  magnetic  flux  per 
pole,  hence  a  lower  supply  voltage  on  the  armature,  and  thus  a 
larger  armature  current  and  therewith  a  larger  commutator,  etc. 

The  insertion  of  reactance  instead  of  resistance  in  the  leads 
connecting  the  commutator  segments  with  the  armature  coils  of 
the  single-phase  motor  also  has  been  proposed  and  used  for 
limiting  the  short-circuit  current  under  the  commutator  brush. 

Reactance  has  the  advantage  over  resistance,  that  the  voltage 

23 


354  ELECTRICAL  APPARATUS 

consumed  by  it  is  wattless  and  therefore  produces  no  serious 
heating  and  reactive  leads  of  low  resistance  thus  are  not  liable 
to  self-destruction  by  heating  if  the  motor  fails  to  start  im- 
mediately. 

On  account  of  the  limited  space  available  in  the  railway  motor 
considerable  difficulty,  however,  is  found  in  designing  sufficiently 
high  reactances  which  do  not  saturate  and  thus  decrease  at 
larger  currents. 

At  speed,  reactance  in  the  armature  coils  is  very  objectionable 
in  retarding  the  reversal  of  current,  and  indeed  one  of  the  most 
important  problems  in  the  design  of  commutating  machines  is  to 
give  the  armature  coils  the  lowest  possible  reactance.  There- 
fore, the  insertion  of  reactance  in  the  motor  leads  interferes 
seriously  with  the  commutation  of  the  motor  at  speed,  and  thus 
requires  the  use  of  a  suitable  commutating  or  reversing  flux,  that 
is,  a  magnetic  field  at  the  commutator  brushes  of  sufficient 
strength  to  reverse  the  current,  against  the  self-inductance  of  the 
armature  coil,  by  means  of  an  e.m.f.  generated  in  the  armature 
coil  by  its  rotation.  This  commutating  flux  thus  must  be  in 
phase  with  the  main  current,  that  is,  a  flux  of  overcompensation. 
Reactive  leads  require  the  use  of  a  commutating  flux  of  over- 
compensation  to  give  fair  commutation  at  speed. 

Counter  E.m.fs.  in  Commutated  Coil 

199.  Theoretically,  the  correct  way  of  eliminating  the  de- 
structive effect  of  the  short-circuit  current  under  the  commu- 
tator brush  resulting  from  the  e.m.f.  of  alternation  of  the  main 
flux  would  be  to  neutralize  the  e.m.f.  of  alternation  by  an  equal 
but  opposite  e.m.f.  inserted  into  the  armature  coil  or  generated 
therein.  Practically,  however,  at  least  with  most  motor  types, 
considerable  difficulty  is  met  in  producing  such  a  neutralizing 
e.m.f.  of  the  proper  intensity  as  well  as  phase.  Since  the  alter- 
nating current  has  not  only  an  intensity  but  also  a  phase  displace- 
ment, with  an  alternating-current  motor  the  production  of  com- 
mutating flux  or  commutating  voltage  is  more  difficult  than  with 
direct-current  motors  in  which  the  intensity  is  the  only  variable. 

By  introducing  an  external  e.m.f.  into  the  short-circuited  coil 
under  the  brush  it  is  not  possible  entirely  to  neutralize  its  e.m.f. 
of  alternation,  but  simply  to  reduce  it  to  one-half.  Several  such 
arrangements  were  developed  in  the  early  days  by  Eickemeyer, 


SINGLE-PHASE  COMMUTATOR  MOTORS        355 


for  instance  the  arrangement  shown  in  Fig.  170,  which  represents 
the  development  of  a  commutator.  The  commutator  consists 
of  alternate  live  segments,  S,  and  dead  segments,  £',  that  is,  seg- 
ments not  connected  to  armature  coils,  and  shown  shaded  in 
Fig.  170.  Two  sets  of  brushes  on  the  commutator,  the  one,  BI, 


I 


s  s 


Bl 


i 


. 


B2 

FIG.  170. — Commutation  with  external  e.m.f. 

ahead  in  position  from  the  other,  B2,  by  one  commutator  seg- 
ment, and  connected  to  the  first  by  a  coil,  N,  containing  an  e.m.f. 
equal  in  phase,  but  half  in  intensity,  and  opposite,  to  the  e.m.f. 
of  alternation  of  the  armature  coil;  that  is,  if  the  armature  coil 
contains  a  single  turn,  coil  N  is  a,  half  turn  located  in  the  main 


m  m 


FIG.  171. — Commutation  by  external  e.m.f. 


m 


field  space;  if  the  armature  coil,  A,  contains  m  turns,  -^  turns  in 

2 

the  main  field  space  are  used  in  coil,  N.  The  dead  segments,  S', 
are  cut  between  the  brushes,  BI  and  B2,  so  as  not  to  short-circuit 
between  the  brushes. 

In  this  manner,  during  the  motion  of  the  brush  over  the  com- 


356  ELECTRICAL  APPARATUS 

mutator,  as  shown  by  Fig.  171  in  its  successive  steps,  in  position: 

1.  There  is  current  through  brush,  BI; 

2.  There  is  current  through  both  brushes,  BI  and  B2,  and  the 

armature  coil,  A,  is  closed  by  the  counter  e.m.f.  of  coil, 
N,  that  is,  the  difference,  A  —  N,  is  short-circuited; 

3.  There  is  current  through  brush  B2; 

4.  There  is  current  through  both  brushes,  BI  and  B2,  and  the 

coil,  N,  is  short-circuited; 

5.  The  current  enters  again  by  brush  BI' 

thus  alternately  the  coil,  N,  of  half  the  voltage  of  the  armature 
coil,  A,  or  the  difference  between  A  and  N  is  short-circuited, 
that  is,  the  short-circuit  current  reduced  to  one-half. 

Complete  elimination  of  the  short-circuit  current  can  be  pro- 
duced by  generating  in  the  armature  coil  an  opposing  e.m.f. 
This  e.m.f.  of  neutralization,  however,  can  not  be  generated  by 
the  alternation  of  the  magnetic  flux  through  the  coil,  as  this  would 
require  a  flux  equal  but  opposite  to  the  full  field  flux  travers- 
ing the  coil,  and  thus  destroy  the  main  field  of  the  motor.  The 
neutralizing  e.m.f.,  therefore,  must  be  generated  by  the  rotation 
of  the  armature  through  the  commutating  field,  and  thus  can 
occur  only  at  speed;  that  is,  neutralization  of  the  short-circuit 
current  is  possible  only  when  the  motor  is  revolving,  but  not  while 
at  rest. 

200.  The  e.m.f.  of  alternation  in  the  armature  coil  short-cir- 
cuited under  the  commutator  brush  is  proportional  to  the  main 
field,  <i>,  to  the  frequency,  /,  and  is  in  quadrature  with  the  main 
field,  being  generated  by  its  rate  of  change;  hence,  it  can  be  rep- 
resented by 

e0  =  27r/$10-8j.  (17) 

The  e.m.f.,  ei,  generated  by  the  rotation  of  the  armature  coil 
through  a  commutating  field,  $',  is,  however,  in  phase  with  the 
field  which  produces  it;  and  since  e\  must  be  equal  and  in  phase 
with  eo  to  neutralize  it,  the  commutating  field,  <£',  therefore,  must 
be  in  phase  with  e0,  hence  in  quadrature  with  3>;  that  is,  the  com- 
mutating field,  <£',  of  the  motor  must  be  in  quadrature  with  the 
main  field,  $,  to  generate  a  neutralizing  voltage,  ei,  of  the  proper 
phase  to  oppose  the  e.m.f.  of  alternation  in  the  short-circuited 
coil.  This  e.m.f.,  ei,  is  proportional  to  its  generating  field,  <£', 
and  to  the  speed,  or  frequency  of  rotation,  /0,  hence  is : 

el  =  27r/o$/10-8,  (18) 


SINGLE-PHASE  COMMUTATOR  MOTORS        357 

and  from  e\  —  e0  it  then  follows  that: 

*'  =  j*  f ;  (19) 

Jo 

that  is,  the  commutating  field  of  the  single-phase  motor  must 
be  in  quadrature  behind  and  proportional  to  the  main  field,  pro- 
portional to  the  frequency  and  inversely  proportional  to  the 
speed;  hence,  at  synchronism,  /0  =  /,  the  commutation  field 
equals  the  main  field  in  intensity,  and,  being  displaced  therefrom 
in  quadrature  both  in  time  and  in  space,  the  motor  thus  must 
have  a  uniform  rotating  field,  just  as  the  induction  motor. 

Above  synchronism,  /0  >  /,  the  commutating  field,  <£',  is  less 
than  the  main  field;  below  synchronism,  however,  /0  <  /,  the 
commutating  field  must  be  greater  than  the  main  field  to  give 
complete  compensation.  It  obviously  is  not  feasible  to  increase 
the  commutating  field  much  beyond  the  main  field,  as  this  would 
require  an  increase  of  the  iron  section  of  the  motor  beyond  that 
required  to  do  the  work,  that  is,  to  carry  the  main  field  flux.  At 
standstill  <£'  should  be  infinitely  large,  that  is,  compensation  is 
not  possible. 

Hence,  by  the  use  of  a  commutating  field  in  time  and  space 
quadrature,  in  the  single-phase  motor  the  short-circuit  current 
under  the  commutator  brushes  resulting  from  the  e.m.f.  of  alter- 
nation can  be  entirely  eliminated  at  and  above  synchronism, 
and  more  or  less  reduced  below  synchronism,  the  more  the  nearer 
the  speed  is  to  synchronism,  but  no  effect  can  be  produced  at 
standstill.  In  such  a  motor  either  some  further  method,  as  re- 
sistance leads,  must  be  used  to  take  care  of  the  short-circuit  cur- 
rent at  standstill,  or  the  motor  designed  so  that  its  commutator 
can  carry  the  short-circuit  current  for  the  small  fraction  of  time 
when  the  motor  is  at  standstill  or  running  at  very  low  speed. 

The  main  field,  <£,  of  the  series  motor  is  approximately  inversely 
proportional  to  the  speed,  /0,  since  the  product  of  speed  and  field 
strength,  /0$,  is  proportional  to  the  e.m.f.  of  rotation,  or  useful 
e.m.f.  of  the  motor,  hence,  neglecting  losses  and  phase  displace- 
ments, to  the  impressed  e.m.f.,  that  is,  constant.  Substituting 

therefore  $  =  7-  <£0,  where  <f>0  =  main  field  at  synchronism,  into 
Jo 

equation  (19) : 

2  (20) 


358 


ELECTRICAL  APPARATUS 


that  is,  the  commutating  field  is  inversely  proportional  to  the 
square  of  the  speed;  for  instance,  at  double  synchronism  it  should 
be  one-quarter  as  high  as  at  synchronism,  etc. 

201.  Of  the  quadrature  field,  3>',  only  that  part  is  needed  for 
commutation  which  enters  and  leaves  the  armature  at  the  posi- 
tion of  the  brushes;  that  is,  instead  of  producing  a  quadrature 
field,  <£',  in  accordance  with  equation  (20),  and  distributed  around 
the  armature  periphery  in  the  same  manner  as  the  main  field,  <f>, 
but  in  quadrature  position  thereto,  a  local  commutating  field 
may  be  used  at  the  brushes,  and  produced  by  a  commutating 
pole  or  commutating  coil,  as  shown  diagrammatically  in  Fig.  172 


FIG.  172. — Commutation  with  commutating  poles. 

as  Ki  and  K.     The  excitation  of  this  commutating  coil,  K,  then 
would  have  to  be  such  as  to  give  a  magnetic  air-gap  density  (B' 
relative  to  that  of  the  main  field,  (B,  by  the  same  equations  (19) 
and  (20) : 


(B'  =  j(B  j 
Jo 


*©• 


(21) 


As  the  alternating  flux  of  a  magnetic  circuit  is  proportional  to 
the  voltage  which  it  consumes,  that  is,  to  the  voltage  impressed 
upon  the  magnetizing  coil,  and  lags  nearly  90°  behind  it,  the  mag- 
netic flux  of  the  commutating  poles,  K,  can  be  produced  by  ener- 
gizing these  poles  by  an  e.m.f.  e,  which  is  varied  with  the  speed 
of  the  motor,  by  equation : 

,  .P  ,      9 

(22) 


where  eQ  is  its  proper  value  at  synchronism. 


SINGLE-PHASE  COMMUTATOR  MOTORS        359 

Since  (B'  lags  90°  behind  its  supply  voltage,  e,  and  also  lags  90° 
behind  (B,  by  equation  (2),  and  so  behind  the  supply  current 
and,  approximately,  the  supply  e.m.f.  of  the  motor,  the  voltage, 
e,  required  for  the  excitation  of  the  commutating  poles  is  approxi- 
mately in  phase  with  the  supply  voltage  of  the  motor;  that  is, 
a  part  thereof  can  be  used,  and  is  varied  with  the  speed  of  the 
motor. 

Perfect  commutation,  however,  requires  not  merely  the  elimi- 
nation of  the  short-circuit  current  under  the  brush,  but  requires 
a  reversal  of  the  load  current  in  the  armature  coil  during  its 
passage  under  the  commutator  brush.  To  reverse  the  current, 
an  e.m.f.  is  required  proportional  but  opposite  to  the  current  and 
therefore  with  the  main  field;  hence,  to  produce  a  reversing  e.m.f. 
in  the  armature  coil  under  the  commutator  brush  a  second  com- 
mutating field  is  required,  in  phase  with  the  main  field  and  ap- 
proximately proportional  thereto. 

The  commutating  field  required  by  a  single-phase  commutator 
motor  to  give  perfect  commutation  thus  consists  of  a  component 
in  quadrature  with  the  main  field,  or  the  neutralizing  component, 
which  eliminates  the  short-circuit  current  under  the  brush,  and 
a  component  in  phase  with  the  main  field,  or  the  reversing  com- 
ponent, which  reverses  the  main  current  in  the  armature  coil 
under  the  brush;  and  the  resultant  commutating  field  thus  must 
lag  behind  the  main  field,  and  so  approximately  behind  the  sup- 
ply voltage,  by  somewhat  less  than  90°,  and  have  an  intensity 
varying  approximately  inversely  proportional  to  the  square  of 
the  speed  of  the  motor. 

Of  the  different  motor  types  discussed  under  IV,  the  series 
motors,  1  and  2,  have  no  quadrature  field,  and  therefore  can  be 
made  to  commutate  satisfactorily  only  by  the  use  of  commutator 
leads,  or  by  the  addition  of  separate  commutating  poles.  The 
inverted  repulsion  motor,  3,  has  a  quadrature  field,  which  de- 
creases with  increase  of  speed,  and  therefore  gives  a  better  com- 
mutation than  the  series  motors,  though  not  perfect,  as  the  quad- 
rature field  does  not  have  quite  the  right  intensity. 

The  repulsion  motors,  4  and  5,  have  a  quadrature  field,  lag- 
ging nearly  90°  behind  the  main  field,  and  thus  give  good  com- 
mutation at  those  speeds  at  which  the  quadrature  field  has  the 
right  intensity  for  commutation.  However,  in  the  repulsion 
motor  with  secondary  excitation,  5,  the  quadrature  field  is  con- 
stant and  independent  of  the  speed,  as  constant  supply  voltage 


360  ELECTRICAL  APPARATUS 

is  impressed  upon  the  commutating  winding,  C,  which  produces 
the  quadrature  field,  and  in  the  direct  repulsion  motor,  4,  the 
quadrature  field  increases  with  the  speed,  as  the  voltage  consumed 
by  the  main  field  F  decreases,  and  that  left  for  the  compensating 
winding,  C,  thus  increases  with  the  speed,  while  to  give  proper 
commutating  flux  it  should  decrease  with  the  square  of  the  speed. 
It  thus  follows  that  the  commutation  of  the  repulsion  motors 
improves  with  increase  of  speed,  up  to  that  speed  where  the 
quadrature  field  is  just  right  for  commutating  field — which  is 
about  at  synchronism — but  above  this  speed  the  commutation 
rapily  becomes  poorer,  due  to  the  quadrature  field  being  far  in 
excess  of  that  required  for  commutating. 

In  the  series  repulsion  motors,  6  and  7,  a  quadrature  field  also 
exists,  just  as  in  the  repulsion  motors,  but  this  quadrature  field 
depends  upon  that  part  of  the  total  voltage  which  is  impressed 
upon  the  commutating  winding,  C,  and  thus  can  be  varied  by 
varying  the  distribution  of  supply  voltage  between  the  two  cir- 
cuits; hence,  in  this  type  of  motor,  the  commutating  flux  can  be 
maintained  through  all  (higher)  speeds  by  impressing  the  total 
voltage  upon  the  compensating  circuit  and  short-circuiting  the 
armature  circuit  for  all  speeds  up  to  that  at  which  the  required 
commutating  flux  has  decreased  to  the  quadrature  flux  given  by 
the  motor,  and  from  this  speed  upward  only  a  part  of  the  supply 
voltage,  inversely  proportional  (approximately)  to  the  square  of 
the  speed,  is  impressed  upon  the  compensating  circuit,  the  rest 
shifted  over  to  the  armature  circuit.  The  difference  between 
6  and  7  is  that  in  6  the  armature  circuit  is  more  inductive,  and 
the  quadrature  flux  therefore  lags  less  behind  the  main  flux  than 
in  7,  and  by  thus  using  more  or  less  of  the  field  coil  in  the  arma- 
ture circuit  its  inductivity  can  be  varied,  and  therewith  the 
phase  displacement  of  the  quadrature  flux  against  the  main  flux 
adjusted  from  nearly  90°  lag  to  considerably  less  lag,  hence  not 
only  the  proper  intensity  but  also  the  exact  phase  of  the  required 
commutating  flux  produced. 

As  seen  herefrom,  the  difference  between  the  different  motor 
types  of  IV  is  essentially  found  in  their  different  actions  regarding 
commutation. 

It  follows  herefrom  that  by  the  selection  of  the  motor-type 
quadrature  fluxes,  <J>i,  can  be  impressed  upon  the  motor,  as  com- 
mutating flux,  of  intensities  and  phase  displacements  against 
the  main  flux,  <J>,  varying  over  a  considerable  range.  The  main 


SINGLE-PHASE  COMMUTATOR  MOTORS        361 

advantage  of  the  series-repulsion  motor  type  is  the  possibility 
which  this  type  affords,  of  securing  the  proper  commutating 
field  at  all  speeds  down  to  that  where  the  speed  is  too  low  to 
induce  sufficient  voltage  of  neutralization  at  the  highest  available 
commutating  flux. 

VI.  Motor  Characteristics 

202.  The  single-phase  commutator  motor  of  varying  speed  or 
series  characteristic  comprises  three  circuits,  the  armature,  the 
compensating  winding,  and  the  field  winding,  which  are  connected 
in  series  with  each  other,  directly  or  indirectly. 

The  impressed  e.m.f .  or  supply  voltage  of  the  motor  then  con- 
sists of  the  components : 

1.  The  e.m.f.  of  rotation,  ei,  or  voltage  generated  in  the  arma- 
ture conductors  by  their  rotation  through  the  magnetic  field,  3>. 
This  voltage  is  in  phase  with  the  field,  $,  and  therefore  approxi- 
mately with  the  current,  i,  that  is,  is  power  e.m.f.,  and  is  the 
voltage  which  does  the  useful  work  of  the  motor.     It  is  propor- 
tional to  the  speed  or  frequency  of  rotation,/0,  to  the  field  strength, 
3>,  and  to  the  number  of  effective  armature  turns,  HI. 

ei  =  27r/0ni$10-8.  (23) 

The  number  of  effective  armature  turns,  Wi,  with  a  distributed 
winding,  is  the  projection  of  all  the  turns  on  their  resultant  direc- 
tion. With  a  full-pitch  winding  of  n  series  turns  from  brush  to 
brush,  the  effective  number  of  turns  thus  is : 

+  ?       2 
HI  =  m[avg  cos]    \  =  -m.  (24) 

-  r      7r 

With  a  fractional-pitch  winding  of  the  pitch  of  r  degrees,  the 
effective  number  of  turns  is : 

m  =  m  -  [avg  cos]    T2  =  -  w  sin  ^'  (25) 

7T  -  —  IT  2t 

2.  The  e.m.f.  of  alternation  of  the  field,  e0,  that  is,  the  voltage 
generated  in  the  field  turns  by  the  alternation  of  the  magnetic 
flux,  <I>,  produced  by  them  and  thus  enclosed  by  them.     This  vol- 
tage is  in  quadrature  with  the  field  flux,  <l>,  and  thus  approxi- 
mately with  the  current  7,  is  proportional  to  the  frequency  of  the 


'X 


362  ELECTRICAL  APPARATUS 


^      impressed  voltage,  /,  to  the  field  strength,  3>,  and  to  the  number  of 
field  turns,  n0. 

e0  =  2J7rfnQ$  10~8.  (26) 

3.  The  impedance  voltage  of  the  motor: 

e'  =  IZ  (27) 

and :  Z  =  r  +  jx, 

where  r  =  total  effective  resistance  of  field  coils,  armature  with 
commutator  and  brushes,  and  compensating  winding,  x  —  total 
self-inductive  reactance,  that  is,  reactance  of  the  leakage  flux  of 
armature  and  compensating  winding — or  the  stray  flux  passing 
locally  between  the  armature  and  the  compensating  conductors 
— plus  the  self-inductive  reactance  of  the  field,  that  is,  the  reac- 
tance due  to  the  stray  field  or  flux  passing  between  field  coils 
and  armature. 

In  addition  hereto,  x  comprises  the  reactance  due  to  the  quad- 
rature magnetic  flux  of  incomplete  compensation  or  overcom- 
pensation,  that  is,  the  voltage  generated  by  the  quadrature  flux, 
$',  in  the  difference  between  armature  and  compensating  con- 
ductors, HI  —  n2  or  n2  —  n\. 

Therefore  the  total  supply  voltage,  E,  of  the  motor  is : 

E  =  61  +  eo  +  e' 

=  2  Tr/o^i*  10~8  +  2  fafniQ  10-8  +  (r  +  jx)  I.     (28) 


Let,  then,  R  =  magnetic  reluctance  of  field  circuit,  thus 
<£  =  -~-  =  the  magnetic  field  flux,  when  assuming  this  flux  as  in 
phase  with  the  excitation  7,  and  denoting: 

*.  (30) 


as  the  effective  reactance  of  field  inductance,  corresponding  to 
the  e.m.f.  of  alternation: 

S  =  j  =  ratio  of  speed  to  frequency,  or  speed 

'        as  fraction  of  synchronism,  , 

(61) 

c  —  --  =  ratio  of  effective  armature  turns  to 
HQ      field  turns; 


SINGLE-PHASE  COMMUTATOR  MOTORS        363 

substituting  (30)  and  (31)  in  (28): 

E  =  cSxQI  +  jxol  +  (r  +  jx)  I 


=  [(r  +  cSxo)  +j(x  +  x0)]f',  (32) 

or: 

E 
I  —  /     i   _o~  '\~i    •  /„   i    „  x'  (33) 

and,  in  absolute  values: 

*  =  ^//y.  •     ™    x!   .    /      •       xo'  (34) 


The  power-factor  is  given  by  : 

tan  e  =   X  +  X"  •  (35) 

r  +  cSx 

The  useful  work  of  the  motor  is  done  by  the  e.m.f  .  of  rotation  : 
E,  =  cSxQI, 

and,  since  this  e.m.f.,  EI,  is  in  phase  with  the  current,  I,  the 
useful  work,  or  the  motor  output  (inclusive  friction,  etc.),  is: 
p  =  EJ  =  cSx0i2 


_  _          _ 
~  (r  +  cSxo)*  +  (x+  so)2' 

and  the  torque  of  the  motor  is  : 
P 

S 

,„„, 


(r 
For  instance,  let  : 

e  =  200  volts,  c  =  —  =  4, 

n0 

Z  =  r  +jx  =  0.02  +  0.06  j,  x0  =  0.08; 


then: 

10,000 


p  =  32,000^ 

(1  +  16  S)2  +  49 

32,000 
D=    (1  +  16  S)2  +  49  Syn' 


364 


ELECTRICAL  APPARATUS 


203.  The  behavior  of  the  motor  at  different  speeds  is  best 
shown  by  plotting  i,  p  =  cos  6,  P  and  D  as  ordinates  with  the 
speed,  S,  as  abscissae,  as  shown  in  Fig.  173. 

In  railway  practice,  by  a  survival  of  the  practice  of  former 
times,  usually  the  constants  are  plotted  with  the  current,  7,  as 
abscissae,  as  shown  in  Fig.  174,  though  obviously  this  arrange- 
ment does  not  as  well  illustrate  the  behavior  of  the  motor. 

Graphically,  by  starting  with  the  current,  7,  as  zero  axis,  07,  the 
motor  diagram  is  plotted  in  Fig.  175. 


FIG.  173. — Single-phase  commutator-motor  speed  characteristics. 


Thevoltage  consumed  by  the  resistance,  r,  is  OEr  =  ir,  in  phase 
with  07;  the  voltage  consumed  by  the  reactance,  x,  is  OEX  =  ix, 
and  90°  ahead  of  07.  OEr  and  OEX  combine  to  the  voltage  con- 
sumed by  the  motor  impedance,  OE'  =  iz. 

Combining  OE'  =  iz,  OEi  =  ei}  and  OEQ  =  e0  thus  gives  the 
terminal  voltage,  OE  =  e,  of  the  motor,  and  the  phase  angle, 

#07  =  e. 

In  this  diagram,  and  in  the  preceding  approximate  calculation, 
the  magnetic  flux,  <£,  has  been  assumed  in  phase  with  the  current,  7. 

In  reality,  however,  the  equivalent  sine  wave  of  magnetic 
flux,  $,  lags  behind  the  equivalent  sine  wave  of  exciting  current,  7, 
by  the  angle  of  hysteresis  lag,  and  still  further  by  the  power 


SINGLE-PHASE  COMMUTATOR  MOTORS 


365 


consumed  by  eddy  currents,  and,  especially  in  the  commutator 
motor,  by  the  power  consumed  in  the  short-circuit  current  under 
the  brushes,  and  the  vector,  0$,  therefore  is  behind  the  current 
vector,  01,  by  an  angle  a,  which  is  small  in  a  motor  in  which  the 
short-circuit  current  under  the  brushes  is  eliminated  and  the 
eddy  currents  are  negligible,  but 'may  reach  considerable  values 
in  the  motor  of  poor  commutation. 


FIG.  174. — Single-phase  commutator-motor  current  characteristics. 

Assuming  then,  in  Fig.  J76,  0$  lagging  behind  01  by  angle 
a,  OEi  is  in  phase  with  0$,  hence  lagging  behind  07;  that  is, 
the  e.m.f.  of  rotation  is  not  entirely  a  power  e.m.f.,  but  contains 
a  wattless  lagging  component.  The  e.m.f.  of  alternation,  OE0, 
is  90°  ahead  of  0$,  hence  less  than  90°  ahead  of  01,  and  therefore 
contains  a  power  component  representing  the  power  consumed 
by  hysteresis,  eddy  currents,  and  the  short-circuit  current  under 
the  brushes. 

Completing  now  the  diagram,  it  is  seen  that  the  phase  angle,  6, 
is  reduced,  that  is,  the  power-factor  of  the  motor  increased  by 


366 


ELECTRICAL  APPARATUS 


the  increased  loss  of  power,  but  is  far  greater  than  corresponding 
thereto.  It  is  the  result  of  the  lag  of  the  e.m.f .  of  rotation,  which 
produces  a  lagging  e.m.f.  component  partially  compensating  for 
the  leading  e.m.f.  consumed  by  self-inductance,  a  lag  of  the  e.m.f. 
being  equivalent  to  a  lead  of  the  current. 


Er 


FIG.  175. — Single-phase  commutator-motor  vector  diagram. 

As  the  result  of  this  feature  of  a  lag  of  the  magnetic  flux,  <£, 
by  producing  a  lagging  e.m.f.  of  rotation  and  thus  compensating 
for  the  lag  of  current  by  self-inductance,  single-phase  motors 
having  poor  commutation  usually  have  better  power-factors,  and 


FIG.  176. — Single-phase  commutator-motor  diagram  with  phase  displace- 
ment between  flux  and  current. 

improvement  in  commutation,  by  eliminating  or  reducing  the 
short-circuit  current  under  the  brush,  usually  causes  a  slight  de- 
crease in  the  power-factor,  by  bringing  the  magnetic  flux,  3>,  more 
nearly  in  phase  with  the  current,  7. 

204.  Inversely,  by  increasing  the  lag  of  the  magnetic  flux,  <E>, 
the  phase  angle  can  be  decreased  and  the  power-factor  improved. 
Such  a  shift  of  the  magnetic  flux,  $,  behind  the  supply  current,  i, 
can  be  produced  by  dividing  the  current,  i,  into  components,  i' 


SINGLE-PHASE  COMMUTATOR  MOTORS        367 


and  i" ,  and  using  the  lagging  component  for  field  excitation. 
This  is  done  most  conveniently  by  shunting  the  field  by  a  non- 
inductive  resistance.  Let  r0  be  the  non-inductive  resistance  in 
shunt  with  the  field  winding,  of  reactance,  XQ  +  x\,  where  x\  is 


FIG.  177. — Single-phase  commutator-motor  improvement  of  power-factor 
by  introduction  of  lagging  e.m.f.  of  rotation. 

that  part  of  the  self-inductive  reactance,  x,  due  to  the  field  coils. 
The  current,  i',  in  the  field  is  lagging  90°  behind  the  current,  i", 
in  a  non-inductive  resistance,  and  the  two  currents  have  the 

;  hence,  dividing  the  total  current,  01,  in  this 


ratio  -77-,  = 

i 


r() 


+  XL 


proportion  into  the  two  quadrature  components,  01'  and  01" ', 


FIG.  178. — Single-phase  commutator  motor.     Unity  power-factor  produced 
by  lagging  e.m.f.  of  rotation. 

in  Fig.  177,  gives  the  magnetic  flux,  0<l>,  in  phase  with  07',  and 
so  lagging  behind  07,  and  then  the  e.m.f.  of  rotation  is  OEi,  the 
e.m.f.  of  alternation  OE0,  and  combining  OEi,  OE0,  and  OE' 


368 


ELECTRICAL  APPARATUS 


gives  the  impressed  e.m.f.;  OE,  nearer  in  phase  to  01  than  with 
0$  in  phase  with  01. 

In  this  manner,  if  the  e.m.fs.  of  self-inductance  are  not  too 
large,  unity  power-factor  can  be  produced,  as  shown  in  Fig.  178. 

Let  01  =  total  current,  OE'  =  impedance  voltage  of  the 
motor,  OE  =  impressed  e.m.f.  or  supply  voltage,  and  assumed 
in  phase  with  01.  OE  then  must  be  the  resultant  of  OE'  and  of 
OE%,  the  voltage  of  rotation  plus  that  of  alternation,  and  resolv- 
ing therefore  OE%  into  two  components,  OEi  and  OE0,  in  quadra- 


FIG.  179. — Single-phase  commutator-motor  diagram  with  secondary 
excitation. 

ture  with  each  other,  and  proportional  respectively  to  the  e.m.f. 
of  rotation  and  the  e.m.f.  of  alternation,  gives  the  magnetic  flux, 
0$,  in  phase  with  the  e.m.f.  of  rotation,  OEi,  and  the  component 
of  current  in  the  field,  07',  and  in  the. non-inductive  resistance, 
01",  in  phase  and  in  quadrature  respectively  with  0$,  which 
combined  make  up  the  total  current.  The  projection  of  the 
e.m.f.  of  rotation  OEi  on  01  then  is  the  power  component  of 
the  e.m.f.,  which  does  the  work  of  the  motor,  and  the  quadra- 
ture projection  of,  OEi,  is  the  compensating  component  of  the 
e.m.f.  of  rotation,  which  neutralizes  the  wattless  component  of 
the  e.m.f.  of  self -inductance. 

Obviously  such  a  compensation  involves  some  loss  of  power 
in  the  non-inductive  resistance,  r0,  shunting  the  field  coils,  and  as 
the  power-factor  of  the  motor  usually  is  sufficiently  high,  such 
compensation  is  rarely  needed. 

In  motors  in  which  some  of  the  circuits  are  connected  inductively 
in  series  with  the  others  the  diagram  is  essentially  the  same,  except 


SINGLE-PHASE  COMMUTATOR  MOTORS        369 

that  a  phase  displacement  exists  between  the  secondary  and  the 
primary  current.  The  secondary  current,  /i,  of  the  transformer 
lags  behind  the  primary  current,  70,  slightly  less  than  180° ;  that  is, 
considered  in  opposite  direction,  the  secondary  current  leads  the 
primary  by  a  small  angle,  0o,  and  in  the  motors  with  secondary 
excitation  the  field  flux,  3>,  being  in  phase  with  the  field  current, 
1 1  (or  lagging  by  angle  a  behind  it),  thus  leads  the  primary 
current,  /o,  by  angle  00  (or  angle  00  —  «) .  As  a  lag  of  the  mag- 
netic flux  3>  increases,  and  a  lead  thus  decreases  the  power-factor, 
motors  with  secondary  field  excitation  usually  have  a  slightly 


FIG.  180. — Single-phase  commutator  motor  with  secondary  excitation 
power-factor  improved  by  shunting  field  winding  with  non-inductive 
circuit. 


lower  power-factor  than  motors  with  primary  field  excitation, 
and  therefore,  where  desired,  the  power-factor  may  be  improved 
by  shunting  the  field  with  a  non-inductive  resistance,  r0.  Thus 
for  instance,  if,  in  Fig.  179,  OI0  =  primary  current,  O/i  =  sec- 
ondary current,  OE1}  in  phase  with  01 \,  is  the  e.m.f.  of  rotation, 
in  the  case  of  the  secondary  field  excitation,  and  OE0,  in  quadra- 
ture ahead  of  O/i,  is  the  e.m.f.  of  alternation,  while  OE'  is  the 
impedance  voltage,  and  OEi,  OEQ  and  OE'  combined  give  the 
supply  voltage,  OE,  and  EOI  =  d  the  angle  of  lag. 

Shunting  the  field  by  a  non-inductive  resistance,  r0,  and  thus 
resolving  the  secondary  current  OIi  into  the  components  OI\  in 
the  field  and  01" \  in  the  non-inductive  resistance,  gives  the  dia- 
gram Fig.  180,  where  a  =  I'i03>  =  angle  of  lag  of  magnetic 
field. 

24 


370  ELECTRICAL  APPARATUS 

205.  The  action  of  the  commutator  in  an  alternating-current 
motor,  in  permitting  compensation  for  phase  displacement  and 
thus  allowing  a  control  of  the  power-factor,  is  very  interesting 
and  important,  and  can  also  be  used  in  other  types  of  machines, 
as  induction  motors  and  alternators,  by  supplying  these  machines 
with  a  commutator  for  phase  control. 

A  lag  of  the  current  is  the  same  as  a  lead  of  the  e.m.f.,  and  in- 
versely a  leading  current  inserted  into  a  circuit  has  the  same  ef- 
fect as  a  lagging  e.m.f.  inserted.  The  commutator,  however, 
produces  an  e.m.f.  in  phase  with  the  current.  Exciting  the  field 
by  a  lagging  current  in  the  field,  a  lagging  e.m.f.  of  rotation  is 
produced  which  is  equivalent  to  a  leading  current.  As  it  is  easy 
to  produce  a  lagging  current  by  self-inductance,  the  commutator 
thus  affords  an  easy  means  of  producing  the  equivalent  of  a 
leading  current.  Therefore,  the  alternating-current  commutator 
is  one  of  the  important  methods  of  compensating  for  lagging 
currents.  Other  methods  are  the  use  of  electrostatic  or  electro- 
lytic condensers  and  of  overexcited  synchronous  machines. 

Based  on  this  principle,  a  number  of  designs  of  induction 
motors  and  other  apparatus  have  been  developed,  using  the 
commutator  for  neutralizing  the  lagging  magnetizing  current 
and  the  lag  caused  by  self-inductance,  and  thereby  producing 
unity  power-factor  or  even  leading  currents.  So  far,  however, 
none  of  them  has  come  into  extended  use. 

This  feature,  however,  explains  the  very  high  power-factors 
feasible  in  single-phase  commutator  motors  even  with  consider- 
able air  gaps,  far  larger  than  feasible  in  induction  motors. 

VII.  Efficiency  and  Losses 

206.  The  losses  in  single-phase  commutator  motors  are  essen- 
tially the  same  as  in  other  types  of  machines: 

(a)  Friction  losses — air  friction  or  windage,  bearing  friction 
and  commutator  brush  friction,  and  also  gear  losses  or  other 
mechanical  transmission  losses. 

(6)  Core  losses,  as  hysteresis  and  eddy  currents.  These  are 
of  two  classes — the  alternating  core  loss,  due  to  the  alternation 
of  the  magnetic  flux  in  the  main  field,  quadrature  field,  and  arma- 
ture and  the  rotating  core  loss,  due  to  the  rotation  of  the  arma- 
ture; through  the  magnetic  field.  The  former  depends  upon  the 
frequency,  the  latter  upon  the  speed. 

(c)  Commutation  losses,  as  the  power  consumed  by  the  short- 


SINGLE-PHASE  COMMUTATOR  MOTORS        371 

circtiit  current  under  the  brush,  by  arcing  and  sparking,  where 
such  exists. 

(d)  izr  losses  in  the  motor  circuits^-the  field  coils,  the  compen- 
sating winding,  the  armature  and  the  brush  contact  resistance. 

(e)  Load  losses,  mainly  represented  by  an  effective  resistance, 
that  is,  an  increase  of  the  total  effective  resistance  of  the  motor 
beyond  the  ohmic  resistance. 

Driving  the  motor  by  mechanical  power  and  with  no  voltage 
on  the  motor  gives  the  friction  and  the  windage  losses,  exclusive 
of  commutator  friction,  if  the  brushes  are  lifted  off  the  commu- 
tator, inclusive,  if  the  brushes  are  on  the  commutator.  Ener- 
gizing now  the  field  by  an  alternating  current  of  the  rated  fre- 
quency, with  the  commutator  brushes  off,  adds  the  core  losses 
to  the  friction  losses;  the  increase  of  the  driving  power  then 
measures  the  rotating  core  loss,  while  a  wattmeter  in  the  field 
exciting  circuit  measures  the  alternating  core  loss. 

Thus  the  alternating  core  loss  is  supplied  by  the  impressed 
electric  power,  the  rotating  core  loss  by  the  mechanical  driving 
power. 

Putting  now  the  brushes  down  on  the  commutator  adds  the 
commutation  losses. 

The  ohmic  resistance  gives  the  i2r  losses,  and  the  difference 
between  the  ohmic  resistance  and  the  effective  resistance,  calcu- 
lated from  wattmeter  readings  with  alternating  current  in  the 
motor  circuits  at  rest  and  with  the  field  unexcited,  represents 
the  load  losses. 

However,  the  different  losses  so  derived  have  to  be  corrected 
for  their  mutual  effect.  For  instance,  the  commutation  losses 
are  increased  by  the  current  in  the  armature;  the  load  losses  are 
less  with  the  field  excited  than  without,  etc. ;  so  that  this  method 
of  separately  determining  the  losses  can  give  only  an  estimate  of 
their  general  magnitude,  but  the  exact  determination  of  the  effi- 
ciency is  best  carried  out  by  measuring  electric  input  and  me- 
chanical output. 

VIII.  Discussion  of  Motor  Types 

207.  Varying-speed  single-phase  commutator  motors  can  be 
divided  into  two  classes,  namely,  compensated  series  motors  and 
repulsion  motors.  In  the  former,  the  main  supply  current  is 
through  the  armature,  while  in  the  latter  the  armature  is, closed 
upon  itself  as  secondary  circuit,  with  the  compensating  winding 


372  ELECTRICAL  APPARATUS 

as  primary  or  supply  circuit.  As  the  result  hereof  the  repulsion 
motors  contain  a  transformer  flux,  in  quadrature  position  to  the 
main  flux,  and  lagging  behind  it,  while  in  the  series  motors  no 
such  lagging  quadrature  flux  exists,  but  in  quadrature  position 
to  the  main  flux,  the  flux  either  is  zero — complete  compensation 
in  phase  with  the  main  flux — over-  or  undercompensation. 


A.  Compensated  Series  Motors 

Series  motors  give  the  best  power-factors,  with  the  exception 
of  those  motors  in  which  by  increasing  the  lag  of  the  field  flux 
a  compensation  for  power-factor  is  produced,  as  discussed  in  V. 
The  commutation  of  the  series  motor,  however,  is  equally  poor 
at  all  speeds,  due  to  the  absence  of  any  commutating  flux,  and 
with  the  exception  of  very  small  sizes  such  motors  therefore  are 
inoperative  without  the  use  of  either  resistance  leads  or  com- 
mutating poles.  With  high-resistance  leads,  however,  fair  opera- 
tion is  secured,  though  obviously  not  of  the  same  class  with 
that  of  the  direct-current  motor ;  with  commutating  poles  or  coils 
producing  a  local  quadrature  flux  at  the  brushes  good  results 
have  been  produced  abroad. 

Of  the  two  types  of  compensation,  conductive  compensation, 
1,  with  the  compensating  winding  connected  in  series  with  the 
armature,  and  inductive  compensation,  2,  with  the  compensated 
winding  short-circuited  upon  itself,  inductive  compensation  nec- 
essarily is  always  complete  or  practically  complete  compensa- 
tion, while  with  conductive  compensation  a  reversing  flux  can 
be  produced  at  the  brushes  by  overcompensation,  and  the  com- 
mutation thus  somewhat  improved,  especially  at  speed,  at  the 
sacrifice,  however,  of  the  power-factor,  which  is  lowered  by  the 
increased  self-inductance  of  the  compensating  winding.  On  the 
short-circuit  current  under  the  brushes,  due  to  the  e.m.f.  of  alter- 
nation, such  overcompensation  obviously  has  no  helpful  effect. 
Inductive  compensation  has  the  advantage  that  the  compen- 
sating winding  is  not  connected  with  the  supply  circuit,  can  be 
made  of  very  low  voltage,  or  even  of  individually  short-circuited 
turns,  and  therefore  larger  conductors  and  less  insulation  used, 
which  results  in  an  economy  of  space,  and  therewith  an  increased 
output  for  the  same  size  of  motor.  Therefore  inductive  compen- 
sation is  preferable  where  it  can  be  used.  It  is  not  permissible, 
however,  in  motors  which  are  required  to  operate  also  on  direct 
current,  since  with  direct-current  supply  no  induction  takes  place 


SINGLE-PHASE  COMMUTATOR  MOTORS        373 

and  therefore  the  compensation  fails,  and  with  the  high  ratio  of 
armature  turns  to  field  turns,  without  compensation,  the  field 
distortion  is  altogether  too  large  to  give  satisfactory  commutation, 
except  in  small  motors. 

The  inductively  compensated  series  motor  with  secondary  ex- 
citation, or  inverted  repulsion  motor,  3,  takes  an  intermediary 
position  between  the  series  motors  and  the  repulsion  motors;  it 
is  a  series  motor  in  so  far  as  the  armature  is  in  the  main  supply 
circuit,  but  magnetically' it  has  repulsion-motor  characteristics, 
that  is,  contains  a  lagging  quadrature  flux.  As  the  field  exci- 
tation consumes  considerable  voltage,  when  supplied  from  the 
compensating  winding  as  secondary  circuit,  considerable  voltage 
must  be  generated  in  this  winding,  thus  giving  a  corresponding 
transformer  flux.  With  increasing  speed  and  therewith  decreas- 
ing current,  the  voltage  consumed  by  the  field  coils  decreases, 
and  therewith  the  transformer  flux  which  generates  this  voltage. 
Therefore,  the  inverted  repulsion  motor  contains  a  transformer 
flux  which  has  approximately  the  intensity  and  the  phase  re- 
quired for  commutation;  it  lags  behind  the  main  flux,  but  less 
than  90°,  thus  contains  a  component  in  phase  with  the  main 
flux,  as  reversing  flux,  and  decreases  with  increase  of  speed. 
Therefore,  the  commutation  of  the  inverted  repulsion  motor  is 
very  good,  far  superior  to  the  ordinary  series  motor,  and  it  can 
be  operated  without  resistance  leads ;  it  has,  however,  the  serious 
objection  of  a  poor  power-factor,  resulting  from  the  lead  of  the 
field  flux  against  the  armature  current,  due  to  the  secondary  ex- 
citation, as  discussed  in  V.  To  make  such  a  motor  satisfactory 
in  power-factor  requires  a  non-inductive  shunt  across  the  field, 
and  thereby  a  waste  of  power.  For  this  reason  it  has  not  come 
into  commercial  use. 

B.  Repulsion  Motors 

208.  Repulsion  motors  are  characterized  by  a  lagging  quadra- 
ture flux,  which  transfers  the  power  from  the  compensating  wind- 
ing to  the  armature.  At  standstill,  and  at  very  low  speeds,  re- 
pulsion motors  and  series  motors  are  equally  unsatisfactory  in 
commutation;  while,  however,  in  the  series  motors  the  commu- 
tation remains  bad  (except  when  using  commutating  devices), 
in  the  repulsion  motors  with  increasing  speed  the  commutation 
rapidly  improves,  and  becomes  perfect  near  synchronism.  As 
the  result  hereof,  under  average  conditions  a  much  inferior  com- 


374  ELECTRICAL  APPARATUS 

mutation  can  be  allowed  in  repulsion  motors  at  very  low  speeds 
than  in  series  motors,  since  in  the  former  the  period  of  poor 
commutation  lasts  only  a  very  short  time.  While,  therefore, 
series  motors  can  not  be  satisfactorily  operated  without  resistance 
leads  (or  commutating  poles),  in  repulsion  motors  resistance 
leads  are  not  necessary  and  not  used,  and  the  excessive  current 
density  under  the  brushes  in  the  moment  of  starting  permitted, 
as  it  lasts  too  short  a  time  to  cause  damage  to  the  commutator. 

As  the  transformer  field  of  the  repulsion  motor  is  approximately 
constant,  while  the  proper  commutating  field  should  decrease 
with  the  square  of  the  speed,  above  synchronism  the  transformer 
field  is  too  large  for  commutation,  and  at  speeds  considerably 
above  synchronism — 50  per  cent,  and  more — the  repulsion  motor 
becomes  inoperative  because  of  excessive  sparking.  At  syn- 
chronism, the  magnetic  field  of  the  repulsion  motor  is  a  rotating 
field,  like  that  of  the  polyphase  induction  motor. 

Where,  therefore,  speeds  far  above  synchronism  are  required, 
the  repulsion  motor  can  not  be  used;  but  where  synchronous 
speed  is  not  much  exceeded  the  repulsion  motor  is  preferred  be- 
cause of  its  superior  commutation.  Thus  when  using  a  commu- 
tator as  auxiliary  device  for  starting  single-phase  induction 
.motors  the  repulsion-motor  type  is  used.  For  high  frequencies, 
as  60  cycles,  where  peripheral  speed  forbids  synchronism  being 
greatly  exceeded,  the  repulsion  motor  is  the  type  to  be  considered. 

Repulsion  motors  also  may  be  built  with  primary  and  sec- 
ondary excitation.  The  latter  usually  gives  a  better  commuta- 
tion, because  of  the  lesser  lag  of  the  transformer  flux,  and  there- 
with a  greater  in-phase  component,  that  is,  greater  reversing  flux, 
especially  at  high  speeds.  Secondary  excitation,  however,  gives 
a  slightly  lower  power-factor. 

A  combination  of  the  repulsion-motor  and  series-motor  types 
is  the  series  repulsion  motor,  6  and  7.  In  this  only  a  part  of 
the  supply  voltage  is  impressed  upon  the  compensating  winding 
and  thus  transformed  to  the  armature,  while  the  rest  of  the  sup- 
ply voltage  is  impressed  directly  upon  the  armature,  just  as  in 
the  series  motor.  As  result  thereof  the  transformer  flux  of  the 
series  repulsion  motor  is  less  than  that  of  the  repulsion  motor, 
in  the  same  proportion  in  which  the  voltage  impressed  upon  the 
compensating  winding  is  less  than  the  total  supply  voltage. 
Such  a  motor,  therefore,  reaches  equality  of  the  transformer  flux 
with  the  commutating  flux,  and  gives  perfect  commutation  at  a 


SINGLE-PHASE  COMMUTATOR  MOTORS        375 

higher  speed  than  the  repulsion  motor,  that  is,  above  synchron- 
ism. With  the  total  supply  voltage  impressed  upon  the  compen- 
sating winding,  the  transformer  flux  equals  the  commutating 
flux  at  synchronism.  At  n  times  synchronous  speed  the  com- 
mutating flux  should  be  — ^  of  what  it  is  at  synchronism,  and  by 

impressing  — ^  of  the  supply  voltage  upon  the  compensating  wind- 

IV 

ing,  the  rest  on  the  armature,  the  transformer  flux  is  reduced 
to  — 2  of  its  value,  that  is,  made  equal  to  the  required  commuta- 

rL 

ting  flux  at  n  times  synchronism. 

In  the  series  repulsion  motor,  by  thus  gradually  shifting  the 
supply  voltage  from  the  compensating  winding  to  the  armature 
and  thereby  reducing  the  transformer  flux,  it  can  be  maintained 
equal  to  the  required  commutating  flux  at  all  speeds  from  syn- 
chronism upward;  that  is,  the  series  repulsion  motor  arrange- 
ment permits  maintaining  the  perfect  commutation,  which  the 
repulsion  motor  has  near  synchronism,  for  all  higher  speeds. 

With  regard  to  construction,  no  essential  difference  exists  be- 
tween the  different  motor  types,  and  any  of  the  types  can  be 
operated  equally  well  on  direct  current  by  connecting  all  three 
circuits  in  series.  In  general,  the  motor  types  having  primary 
and  secondary  circuits,  as  the  repulsion  and  the  series  repulsion 
motors,  give  a  greater  flexibility,  as  they  permit  winding  the 
circuits  for  different  voltages,  that  is,  introducing  a  ratio  of  trans- 
formation between  primary  and  secondary  circuit.  Shifting  one 
motor  element  from  primary  to  secondary,  or  inversely,  then 
gives  the  equivalent  of  a  change  of  voltage  or  change  of  turns, 
Thus  a  repulsion  motor  in  which  the  stator  is  wound  for  a  higher 
voltage,  that  is,  with  more  turns,  than  the  rotor  or  armature, 
when  connecting  all  the  circuits  in  series  for  direct-current  opera- 
tion, gives  a  direct-current  motor  having  a  greater  field  excita- 
tion compared  with  the  armature  reaction,  that  is,  the  stronger 
field  which  is  desirable  for  direct-current  operating  but  not  per- 
missible with  alternating  current. 

209.  In  general,  tthe  construct ve  differences  between  motor 
types  are  mainly  differences  in  connection  of  the  three  circuits. 
For  instacne,  let  F  =  field  circuit,  A  =  armature  circuit,  C  = 
compensating  circuit,  T  =  supply  transformer,  R  —  resistance 
used  in  starting  and  at  very  low  speeds.  Connecting,  in  Fig.  181, 
the  armature,  A,  between  field  F  and  compensating  winding,  C. 


376 


ELECTRICAL  APPARATUS 


With  switch  0  open  the  starting  resistance  is  in  circuit;  closing 
switch  0  short-circuits  the  starting  resistance  and  gives  the  run- 
ning conditions  of  the  motor. 

With  all  the  other  switches  open  the  motor  is  a  conductively 
compensated  series  motor. 


F  A  C 

FIG.  181. — Alternating-current   commutator  motor   arranged   to   operate 
either  as  series  or  repulsion  motor. 

Closing  1  gives  the  inductively  compensated  series  motor. 
Closing  2  gives  the  repulsion  motor  with  primary  excitation. 
Closing  3  gives  the  repulsion  motor  with  secondary  excitation . 
Closing  4  or  5  or  6  or  7  gives  the  successive  speed  steps  of  the 
series  repulsion  motor  with  armature  excitation. 


fi 

4)          5) 

r     r 

6)          7) 

r     r 

f 

^      Va  — 

;# 

In/ 

1 

y 

( 

in 

FIG.  182. — Alternating-current    commutator   motor   arranged   to    operate 
either  as  series  or  repulsion  motor. 

Connecting,  in  Fig.  182,  the  field,  F,  between  armature,  A,  and 
compensating  winding,  C,  the  resistance,  R,  is  again  controlled  by 
switch  0. 

All  other  switches  open  gives  the  conductively  compensated 
series  motor. 


SINGLE-PHASE  COMMUTATOR  MOTORS        377 

Switch  1  closed  gives  the  inductively  compensated  series 
motor. 

Switch  2  closed  gives  the  inductively  compensated  series 
motor  with  secondary  excitation,  or  inverted  repulsion  motor. 

Switch  3  closed  gives  the  repulsion  motor  with  primary 
excitation. 

Switches  4  to  7  give  the  different  speed  steps  of  the  series  re- 
pulsion motor  with  primary  excitation. 

Opening  the  connection  at  x  and  closing  at  y  (as  shown  in 
dotted  line),  the  steps  3  to  7  give  respectively  the  repulsion  motor 
with  secondary  excitation  and  the  successive  steps  of  the  series 
repulsion  motor  with  armature  excitation. 

Still  further  combinations  can  be  produced  in  this  manner,  as 
for  instance,  in  Fig.  181,  by  closing  2  and  4,  but  leaving  0  open, 
the  field,  F,  is  connected  across  a  constant-potential  supply,  in 
series  with  resistance,  R,  while  the  armature  also  receives  con- 
stant voltage,  and  the  motor  then  approaches  a  finite  speed,  that 
is,  has  shunt  motor  characteristic,  and  in  starting,  the  main 
field,  F,  and  the  quadrature  field,  AC,  are  displaced  in  phase,  so 
give  a  rotating  or  polyphase  field  (unsymmetrical) . 

To  discuss  all  these  motor  types  with  their  in  some  instances 
very  interesting  characteristics  obviously  is  not  feasible.  In 
general,  they  can  all  be  classified  under  series  motor,  repulsion 
motor,  shunt  motor,  and  polyphase  induction  motor,  and  com- 
binations thereof. 

IX.  Other  Commutator  Motors 

210.  Single-phase  commutator  motors  have  been  developed  as 
varying-speed  motors  for  railway  service.  In  other  directions 
commutators  have  been  applied  to  alternating-current  motors 
and  such  motors  developed: 

(a)  For  limited  speed,  or  of  the  shunt-motor  type,  that  is, 
motors  of  similar  characteristic  as  the  single-phase  railway 
motor,  except  that  the  speed  does  not  indefinitely  increase  with 
decreasing  load  but  approaches  a  finite  no-load  value.  Several 
types  of  such  motors  have  been  developed,  as  stationary  motors 
for  elevators,  variable-speed  machinery,  etc.,  usually  of  the 
single-phase  type. 

By  impressing  constant  voltage  upon  the  field  the  magnetic 
field  flux  is  constant,  and  the  speed  thus  reaches  a  finite  limiting 
value  at  which  the  e.m.f.  of  rotation  of  the  armature  through 


378  ELECTRICAL  APPARATUS 

the  constant  field  flux  consumes  the  impressed  voltage  of  the 
armature.  By  changing  the  voltage  supply  to  the  field  different 
speeds  can  be  produced,  that  is,  an  adjustable-speed  motor. 
The  main  problem  in  the  design  of  such  motors  is  to  get  the 
field  excitation  in  phase  with  the  armature  current  and  thus  pro- 
duce a  good  power-factor. 

(6)  Adjustable-speed  polyphase  induction  motors.  In  the 
secondary  of  the  polyphase  induction  motor  an  e.m.f.  is  gener- 
ated which,  at  constant  impressed  e.m.f.  and  therefore  approxi- 
mately constant  flux,  is  proportional  to  the  slip  from  synchron- 
ism. With  short-circuited  secondary  the  motor  closely  ap- 
proaches synchronism.  Inserting  resistance  into  the  secondary 
reduces  the  speed  by  the  voltage  consumed  in  the  secondary. 
As  this  is  proportional  to  the  current  and  thus  to  the  load,  the 
speed  control  of  the  polyphase  induction  motor  by  resistance  in 
the  secondary  gives  a  speed  which  varies  with  the  load,  just  as 
the  speed  control  of  a  direct-current  motor  by  resistance  in  the 
armature  circuit ;  hence,  the  speed  is  not  constant,  and  the  opera- 
tion at  lower  speeds  inefficient.  Inserting,  however,  a  constant 
voltage  into  the  secondary  of  the  induction  motor  the  speed  is 
decreased  if  this  voltage  is  in  opposition,  and  is  increased  if  this 
voltage  is  in  the  same  direction  as  the  secondary  generated  e.m.f., 
and  in  this  manner  a  speed  control  can  be  produced.  If  c  = 
voltage  inserted  into  the  secondary,  as  fraction  of  the  voltage 
which  would  be  induced  in  it  at  full  frequency  by  the  rotating 
field,  then  the  polyphase  induction  motor  approaches  at  no-load 
and  runs  at  load  near  to  the  speed  (1  —  c)  or  (1  +  c)  times  syn- 
chronism, depending  upon  the  direction  of  the  inserted  voltage. 

Such  a  voltage  inserted  into  the  induction-motor  secondary 
must,  however,  have  the  frequency  of  the  motor  secondary  cur- 
rents, that  is,  of  slip,  and  therefore  can  be  derived  from  the  full- 
frequency  supply  circuit  only  by  a  commutator  revolving  with 
the  secondary.  If  cf  is  the  frequency  of  slip,  then  (1  —  c)/  is 
the  frequency  of  rotation,  and  thus  the  frequency  of  commuta- 
tion, and  at  frequency,  /,  impressed  upon  the  commutator  the 
effective  frequency  of  the  commutated  current  is/  —  (I  —  c)  f  = 
cf,  or  the  frequency  of  slip,  as  required. 

Thus  the  commutator  affords  a  means  of  inserting  voltage 
into  the  secondary  of  induction  motors  and  thus  varying  its 
speed. 

However,  while  these  commutated  currents  in  their  resultant 


SINGLE-PHASE  COMMUTATOR  MOTORS         379 

give  the  effect  of  the  frequency  of  slip,  they  actually  consist  of 
sections  of  waves  of  full  frequency,  that  is,  meet  the  full  station- 
ary impedance  in  the  rotor  secondary,  and  not  the  very  much 
lower  impedance  of  the  low-frequency  currents  in  the  ordinary 
induction  motor. 

If,  therefore,  the  brushes  on  the  commutator  are  set  so  that 
the  inserted  voltage  is  in  phase  with  the  voltage  generated  in  the 
secondary,  the  power-factor  of  the  motor  is  very  poor.  Shifting 
the  brushes,  by  a  phase  displacement  between  the  generated  and 
the  inserted  voltage,  the  secondary  currents  can  be  made  to  lead, 
and  thereby  compensate  for  the  lag  due  to  self-inductance  and 
unity  power-factor  produced.  This,  however,  is  the  case  only 
at  one  definite  load,  and  at  all  other  loads  either  overcompensa- 
tion  or  undercompensation  takes  place,  resulting  in  poor  power- 
factor,  either  lagging  or  leading.  Such  a  polyphase  adjustable- 
speed  motor  thus  requires  shifting  of  the  brushes  with  the  load 
or  other  adjustment,  to  maintain  reasonable  power-factor,  and 
for  this  reason  has  not  been  used. 

(c)  Power-factor  compensation.  The  production  of  an  alter- 
nating magnetic  flux  requires  wattless  or  reactive  volt-amperes, 
which  are  proportional  to  the  frequency.  Exciting  an  induction 
motor  not  by  the  stationary  primary  but  by  the  revolving  sec- 
ondary, which  has  the  much  lower  frequency  of  slip,  reduces  the 
volt-amperes  excitation  in  the  proportion  of  full  frequency  to 
frequency  of  slip,  that  is,  to  practically  nothing.  This  can  be  done 
by  feeding  the  exciting  current  into  the  secondary  by  commuta- 
tor. If  the  secondary  contains  no  other  winding  but  that  con- 
nected to  the  commutator,  the  motor  gives  a  poor  power-factor. 
If,  however,  in  addition  to  the  exciting  winding,  fed  by  the  com- 
mutator, a  permanently  short-circuited  winding  is  used,  as  a 
squirrel-cage  winding,  the  exciting  impedance  of  the  former  is 
reduced  to  practically  nothing  by  the  short-circuit  winding  coin- 
cident with  it,  and  so  by  overexcitation  unity  power-factor  or 
even  leading  current  can  be  produced.  The  presence  of  the  short- 
circuited  winding,  however,  excludes  this  method  from  speed 
control,  and  such  a  motor  (Heyland  motor)  runs  near  synchron- 
ism just  as  the  ordinary  induction  motor,  differing  merely  by  the 
power-factor.  Regarding  hereto  see  Chapter  on  "  Induction 
Motors  with  Secondary  Excitation/' 

This  method  of  excitation  by  feeding  the  alternating  current 
through  a  commutator  into  the  rotor  has  been  used  very  success- 


380 


ELECTRICAL  APPARATUS 


fully  abroad  in  the  so-called  "compensated  repulsion  motor"  of 
Winter-Eichberg.  This  motor  differs  from  the  ordinary  repul- 
sion motor  merely  by  the  field  coil,  F,  in  Fig.  183  being  replaced 
by  a  set  of  exciting  brushes,  G,  in  Fig.  184,  at  right  angles  to  the 
main  brushes  of  the  armature,  that  is,  located  so  that  the  m.m.f. 
of  the  current  between  the  brushes,  G,  magnetizes  in  the  same 


FIG.  183. — Plain  repulsion  motor. 

direction  as  the  field  coils,  F,  in  Fig.  183.  Usually  the  exciting 
brushes  are  supplied  by  a  transformer  or  autotransformer,  so  as 
to  vary  the  excitation  and  thereby  the  speed. 

This  arrangement  then  lowers  the  e.m.f.  of  self-inductance  of 
field  excitation  of  the  motor  from  that  corresponding  to  full  fre- 


FIG.  184. — Winter-Eichberg  motor. 

quency  in  the  ordinary  repulsion  motor  to  that  of  the  frequency 
of  slip,  hence  to  a  negative  value  above  synchronism;  so  that 
hereby  a  compensation  for  lagging  current  can  be  produced 
above  synchronism,  and  unity  power-factor  or  even  leading 
currents  produced. 


SINGLE-PHASE  COMMUTATOR  MOTORS        381 


211.  Theoretical  Investigation. — In  its  most  general  form,  the 
single-phase  commutator  motor,  as  represented  by  Fig.  185, 
comprises:  two  armature  or  rotor  circuits  in  quadrature  with 
each  other,  the  main,  or  energy,  and  the  exciting  circuit  of  the 
armature  where  such  exists,  which  by  a  multisegmental  commu- 
tator are  connected  to  two  sets  of  brushes  in  quadrature  position 
with  each  other.  These  give  rise  to  two  short-circuits,  also  in 
quadrature  position  with  each  other  and  caused  respectively  by 
the  main  and  by  the  exciting  brushes.  Two  stator  circuits,  the 

I2 


I. 


FIG.  185. 

field,  or  exciting,  and  the  cross,  or  compensating  circuit,  also  in 
quadrature  with  each  other,  and  in  line  respectively  with,  the 
exciting  and  the  main,  armature  circuit. 

These  circuits  may  be  separate,  or  may  be  parts  or  components 
of  the  same  circuit.  They  may  be  massed  together1  in  a  single 
slot  of  the  magnetic  structure,  or  may  be  distributed  over  the 
whole  periphery,  as  frequently  done  with  the  armature  windings, 
and  then  as  their  effective  number  of  turns  must  be  considered 
their  vector  resultant,  that  is: 

n  =  -  n' ; 

7T 

where  n'  =  actual  number  of  turns  in  series  between  the  arma- 
ture brushes,  and  distributed  over  the  whole  periphery,  that  is, 
an  arc  of  1 80°  electrical.  Or  the  windings  of  the  circuit  may  be 
distributed  only  over  an  arc  of  the  periphery  of  angle,  co,  as 
frequently  the  case  with  the  compensating  winding  distributed 
in  the  pole  face  of  pole  arc,  co;  or  with  fractional-pitch  armature 
windings  of  pitch,  co.  In  this  case,  the  effective  number  of  turns 

is:  2     .   .     co 

n  =  -^  n'  sin  - 
co  2 


382  ELECTRICAL  APPARATUS 

where  n'  with  a  fractional-pitch  armature  winding  is  the  number 
of  series  turns  in  the  pitch  angle,  co,  that  is: 


n'  = 

7T 

n"  being  the  number  of  turns  in  series  between  the  brushes,  since 
in  the  space  (TT  —  w)  outside  of  the  pitch  angle  the  armature 
conductors  neutralize  each  other,  that  is,  conductors  carrying 
current  in  opposite  direction  are  superposed  upon  each  other. 
See  fractional-pitch  windings,  chapter  "Commutating  Machine," 
"  Theoretical  Elements  of  Electrical  Engineering." 

212.  Let: 

EQ,  (I0,  ZQ  =  impressed  voltage,  current  and  self-inductive 
impedance  of  the  magnetizing  or  exciter  circuit  of  stator  (field 
coils),  reduced  to  the  rotor  energy  circuit  by  the  ratio  of  effective 
turns,  Co, 

Ei,  7i,  Zi  =  impressed  voltage,  current  and  self-inductive  im- 
pedance of  the  rotor  energy  circuit  (or  circuit  at  right  angles 
to  70), 

7?2,  72,  Z2  =  impressed  voltage,  current  and  self -inductive  im- 
pedance of  the  stator  compensating  circuit  (or  circuit  parallel  to 
7i)  reduced  to  the  rotor  circuit  by  the  ratio  of  effective  turns,  c2. 

Es,  I s,  Zi  =  impressed  voltage,  current Jand  self-inductive  im- 
pedance of  the  exciting  circuit  of  the  rotor,  or  circuit  parallel 
to  70, 

7  4,  Z4  =  current  and  self -inductive  impedance  of  the  short- 
circuit  under  the  brushes,  7i,  reduced  to  the  rotor  circuit, 

7  5,  Z5  =  current  and  self-inductive  impedance  of  the  short- 
circuit  under  the  brushes,  73,  reduced  to  the  rotor  circuit, 

Z  =  mutual  impedance  of  field  excitation,  that  is,  in  the  direc- 
tion of  70,  7  3,  7  4, 

Zf  =  mutual  impedance  of  armature  reaction,  that  is,  in  the 
direction  of  7i,  72,  75. 

Z'  usually  either  equals  Z,  or  is  smaller  than  Z. 

74  and  75  are  very  small,  Z4  and  Z5  very  large  quantities. 

Let  S  =  speed,  as  fraction  of  synchronism. 

Using  then  the  general  equations  7  Chapter  XIX,  which  apply 
to  any  alternating-current  circuit  revolving  with  speed,  S,  through 
a  magnetic  field  energized  by  alternating-current  circuits,  gives 
for  the  six  circuits  of  the  general  single-phase  commutator  motor 
the  six  equations: 


SINGLE-PHASE  COMMUTATOR  MOTORS  383 

E0  =  Zo7o  +  Z  (7o  +  73  -  A),  (1) 

E,    =    Z!/!   +   Z'  (I,   +   75    -    /,)    -  JSZ    (/0   +   /3    -    74),,  (2) 

#2  =  Z2/2  +  Z'  (h  -  7i  -  A),  (3) 

#3  =  Z!/3  +  Z  (/,  +  7o  -  74)  -  JSZ  (h  -  7i  -  A),  (4) 

o  =  Z4/4  +  Z  (74  -  /o  -  /«)  -  JSZ  (/!  +  75  -  72),  (5) 

o  =  Z5/5  +  Z'  (75  +  /i  -  /2)  -  JSZ  (/o  +  /3  -  /4)  .  (6) 


These  six  equations  contain  ten  variables: 

/O,    7lj    72,    7s,    74,    7s,    ^0,    $1,    $2,    ^3, 

and  so  leave  four  independent  variables,  that  is,  four  conditions, 
which  may  be  chosen. 

Properly  choosing  these  four  conditions,  and  substituting  them 
into  the  six  equations  (1)  to  (6),  so  determines  all  ten  variables. 
That  is,  the  equations  of  practically  all  single-phase  commutator 
motors  are  contained  as  special  cases  in  above  equations,  and 
derived  therefrom,  by  substituting  the  four  conditions,  which 
characterize  the  motor. 

Let  then,  in  the  following,  the  reduction  factors  to  the  arma- 
ture circuit,  or  the  ratio  of  effective  turns  of  a  circuit,  i,  to  the 
effective  turns  of  the  armature  circuit,  be  represented  by  d. 
That  is, 

number  of  effective  turns  of  circuit,  i 
Cl  ~  number  of  effective  turns  of  armature  circuit' 

and  if  E{,  Ii}  Z{  are  voltage,  current  and  impedance  of  circuit,  i, 
reduced  to  the  armature  circuit,  then  the  actual  voltage,  current 
and  impedance  of  circuit,  i,  are: 


213.  The  different  forms  of  single-phase  commutator  motors, 
of  series  characteristic  are,  as  shown  diagrammatically  in  Fig. 
186: 

1.  Series  motor: 

e  =  CQ&O  +  Ei;  7o  =  Co7iJ  ^2  =  0;  73  =  0. 

2.  Conductively    compensated     series     motor    (Eickemeyer 
motor)  : 

e  =  c0EQ  +  Ei  +  c2#2;  7o  =  Co7i;  h  =  c27i;  h  =  0. 

3.  Inductively  compensated  series  motor  (Eickemeyer  motor)  : 

e  =  c0#o  +  #1;  E2  =  0;  7o  =  c07ij  h  =  0. 


384 


ELECTRICAL  APPARATUS 


4.  Inverted  repulsion  motor,  or  series  motor  with  secondary 
excitation  : 


e  =  Ei',  c0#o  +  c2E2  =  0;  c2/0  =  c0/2;  h  =  0. 

5.  Repulsion  motor  (Thomson  motor)  : 

e  =  cQEQ  -f  c2E2',  Ei  =  0;  c2/0  =  c0/2;  /3  =  0. 

6.  Repulsion  motor  with  secondary  excitation: 

e  =  CzEz',  coE'o  +  EI  =  0;  70  =  c0/i;  73  =  0. 


AO 


(2) 


(3J_ 
^ 

AO 


I 


(61 

X 

AO 


(7) 


AO 


— -n 

Z) 


FIG.  186. 

7.  Series  repulsion  motor  with  secondary  excitation: 

ei  =  c0E0  +  Ei-  e2  =  E2;  h[=  c0/ij  h  =  0. 

8.  Series  repulsion  motor  with  primary  excitation  (Alexander- 
sen  motor)  : 

ei  =  Ei',  e2  =  CoE0  +  c2Ez;  c2/0  =  c0/2;  73  =  0. 

9.  Compensated    repulsion    motor    (Winter    and    Eichberg 
motor)  : 


e  =  c2E2  + 


=  0;  Jo  =  0;  c3/2  =  c2/3. 


SINGLE-PHASE  COMMUTATOR  MOTORS        385 

10.  Rotor-excited  series  motor  with  conductive  compensation: 

e  =  E!  +  c2#2  +  c3#3;  h  =  c2/ij  /3  =  c3/i;  70  =  0. 

11.  Rotor-excited  series  motor  with  inductive  compensation: 

e  =  E,  +  c3#3;  &  =  0;  70  =  0;  73  =  c8/i. 

Numerous  other  combinations  can  be  made  and  have  been 
proposed. 

All  of  these  motors  have  series  characteristics,  that  is,  a  speed 
increasing  with  decrease  of  load.  '  _ 

(1)  to  (8)  contain  only  one  set  of 
brushes  on  the  armature;  (9)  to  (11) 
two  sets  of  brushes  in  quadrature. 

Motors  with  shunt  characteristic, 
that  is,  a  speed  which  does  not  vary 
greatly  with  the  load,  and  reaches  such  FIQ  187 

a  definite  limiting   value    at    no-load 

xthat  the  motor  can  be  considered  a  constant-speed  motor,  can 
also  be  derived  from  the  above  equations.     For  instance: 

Compensated  shunt  motor  (Fig.  187)  : 

El  =  0;  c2E2  =  c3#3  =  e;  /0  =  0. 


In  general,  a  series  characteristic  results,  if  the  field-exciting 
circuit  and  the  armature  energy  circuit  are  connected  in  series 
with  each  other  directly  or  inductively,  or  related  to  each  other 
so  that  the  currents  in  the  two  circuits  are  more  or  less  propor- 
tional to  each  other.  Shunt  characteristic  fesults,  if  the  voltage 
impressed  upon  the  armature  energy  circuit,  and  the  field  excita- 
tion, or  rather  the  magnetic  field  flux,  whether  produced  or  in- 
duced by  the  internal  reactions  of  the-  /motor,  are  constant,  or, 
more  generally,  proportional  to  each  other. 

Repulsion  Motor 

As  illustration  of  the  application  of  these  general  equations, 
paragraph  212,  may  be  considered  the  theory  of  the  repulsion 
motor  (5),  in  Fig.  186. 

214.  Assuming  in  the  following  the  armature  of  the  repulsion 
motor  as  short-circuited  upon  itself,  and  applying  to  the  motor 
the  equations  (1)  to  (6),  the  four  conditions  characteristic  of  the 
repulsion  motor  are: 

25 


386  ELECTRICAL  APPARATUS 

1.  Armature  short-circuited  upon  itself.     Hence: 

#1  =  0. 

2.  Field  circuit  and  cross-circuit  in  series  with  each  other  con- 
nected to  a  source  of  impressed  voltage,  e.     Hence,  assuming 
the  compensating  circuit  or  cross-circuit  of  the  same  number  of 
effective  turns  as  the  rotor  circuit,  or,  c2  =  1  : 


2  =  e. 
Herefrom  follows:  • 

3.    /O    =    Co/2. 

4.  No  armature  excitation  used,  but  only  one  set  of  commu- 
tator brushes;  hence: 

/•=0, 

and  therefore: 


Substituting  these  four  conditions  in  the  six  equations  (1)  to 
(6),  gives  the  three  repulsion  motor  equations: 
Primary  circuit: 

Z2/2  +  Z'  (72  -  /O  +  Co2Zo/2  +  c0Z  (co/2  -  I*)  =  e;      (7) 
Secondary  circuit: 

ZJi  +  Z'  (/!  -  72)  -  JSZ  (c0/2  -  /4)  =  0;  (8) 

Brush  short-circuit: 

Substituting  now  the  abbreviations: 

Z2  +  co2  (Z0  +  Z)  =  Z3,  (10) 

§  =  A,  (11) 

— l  -  \  ^19^ 

z,   -     Ai,  U^ 

"^     ~~~v  —  X4;  (13) 


where  Xi  and  \4,  especially  the  former,  are  small  quantities. 
From  (9)  then  follows: 

/4  =  X4  {/2  (co  -  J5A)  +  jSfrA } ;  (14) 


SINGLE-PHASE  COMMUTATOR  MOTORS        387 
from  (8)  follows,  by  substituting  (14)  and  rearranging: 

1  +T°~  ^r  +  T/  (is) 

1  +  Xi  -  X4S2 
and,  substituting  (15)  in  (14),  gives: 

/4  =  X4/2  1  +  X    —  X  ~S~*~ 

or,  canceling  terms  of  secondary  order  in  the  numerator: 


i  +  Xl  _  x 
Equation  (7)  gives,  substituting  (10)  and  rearranging: 

72  (Z,  +  Zr)  -  7^'  -  I*cQZ  =  e.  (17) 

Substituting  (15)  and  (16)  herein,  and  rearranging,  gives: 
Primary  Current: 


where  : 

K  =  (A3-^co)+Xi(A3+A)-X4(>S2A3-iS2Co+co2-j>Sco),    (19) 
and: 

A,  =  |3;  (20) 

or,  since  approximately: 

A  3  =  Co2,  (21) 

it  is: 

,  K  =  (A  3  -  j^Sco)  +  \i  (co2  -f  A)  -  X4c0  (c0  -  jS).      (22) 


(23) 


(24) 


Substituting  (18),  (19),  (20)  in  (15)  and  (16),  gives: 
Secondary  Current: 


ZK 

Brush  Short-circuit  Current: 


388  ELECTRICAL  APPARATUS 

As  seen,  for  S  =  1,  or  at  synchronism,  74  =  0,  that  is,  the 
short-circuit  current  under  the  commutator  brushes  of  the  re- 
pulsion motor  disappears  at  synchronism,  as  was  to  be  expected, 
since  the  armature  coils  revolve  synchronously  in  a  rotating  field. 

215.  The  e.m.f.  of  rotation,  that  is,  the  e.m.f .  generated  in  the 
rotor  by  its  rotation  through  the  magnetic  field,  which  e.m.f., 
with  the  current  in  the  respective  circuit,  produces  the  torque 
and  so  gives  the  power  developed  by  the  motor,  is : 

Main  circuit: 

E\  =  JSZ  (co/2  -  /4).  (25) 

Brush  short-circuit: 

#'4  =  JSZ'  (7i  -  /,).  (26) 

Substituting  (18),  (23),  (24)  into  (25)  and  (26),  and  rearrang- 
ing, gives: 

Main  Circuit  E.m.f.  of  Rotation: 

l+Xi-M.  (27) 


Brush  Short-circuit  E.m.f.  of  Rotation: 

E\  =  ~  {ScQ  +  jXiA  —  c0X4) ;  (28) 

or,  neglecting  smaller  terms: 

ni  A^     COv  /  *-krf-\\ 

^4    =    — 5v  (29) 

The  Power  produced  by  the  main  armature  circuit  is: 
hence,  substituting  (22)  and  (27) : 


Let: 

m  =  [ZK]  (31) 

be  the  absolute  value  of  the  complex  product,  ZK,  and: 


=  < 

X!  =  Xr,  - 
X4  =  X'4  + 


(32) 


SINGLE-PHASE  COMMUTATOR  MOTORS        389 
it  is,  substituting  (31),  (32)  in  (30),  and  expanding: 

P,  =  {[x(l  -  Scoa")  -  rScoa']  +  (1  -  Sc0r")  [x(\'i  -  X'4) 


-  r(X"i  +  X"4)]  -  Sc0a'[r(X'i  -  X'4)  +x(\"l  +  X"4)] 
-  x  (X'4S2  -  X'4Sc0a"  +  X"4Sc0a')  +  r  (X'4Sc0a' 

},  (33) 


after  canceling  terms  of  secondary  order. 
As  first  approximation  follows  herefrom: 

Sc0e*x 


c0e*x/         „      „       r  „      f\ 
1  i  =  --  5—  (  1  —  oCoo:     --  oCoa   I 
m^    \  xl 


/I       I       C2\ 
Co  (1    +    O2) 

hence  a  maximum  for  the  speed  S,  given  by: 

*S-0 

dS  ' 

or: 

-S«  =  ^1  +  Co*  (a"  +  £  a')  2   -  Co  (a"  +  ^  «').         (35) 
and  equal  to: 


c°2  °"  +  a'  •  c°  °"  +  ~  a/  •  (36) 


The  complete  expression  of  the  power  of  the  main  circuit  is, 
from  (33): 

Pi  =  —  °-      l  -  Sc0    0"  +  -«'!  -60  -  feiS  -  62S2,      (37) 


where  60,  &i,  ^2  are  functions  of  X'i,  X"i,  X'4,  X"4,  as  derived  by 
rearranging  (33). 

The  Power  produced  in  the  brush  short-circuit  is: 


390  ELECTRICAL  APPARATUS 

hence,  substituting  (24)  and  (28)  : 

_  |-£2c0e  X4ec0(l  -  £2)y 

^4  ~  L  x  '     ~z#      J 


(3g) 


hence  positive,  or  assisting,  below  synchronism,  retarding  above 
synchronism. 

The  total  Power,  or  Output  of  the  motor  then  is: 


or: 

Power  Output: 


P  =  Pi  +  P* 


IT-  /  f  \    ~\  I~          /  iv  \  T 

P/^uv  •")  I .,    Q   /  //  i  *    /\  i    T,  i  a  i    /\ //   i^>/\    7_n 
— ^—       1  — oCo  la    H —  a  )     —  Oo  +  o     Co  I  X   4  H —  X  4)  —  61 
IL  \  iC/J  L\  xi  J 

-^2-^Co(x-4  +  ^V4)j;       (39) 


or,  approximately: 


/SCo62X 

m'- 


p  =  ^L^  M  -  Scfl  (a"  +  ^  a')  j 


Se*x\l  -  Sc0(  a"  +  -<*'}  } 

1  »  X       /  J 


(41) 


cod 

hence: 
Torque: 


given  in  synchronous  watts. 

The  power  input  into  the  motor,  and  the  volt-ampere  input, 
are,  if: 


and: 


(42) 


given  by: 
Power  Input: 

PQ  =  ei't,  (43) 


SINGLE-PHASE  COMMUTATOR  MOTORS        391 

Volt-ampere  Input: 

Pa0  =  ei2,  (44) 

Power-factor: 

p  =  ^,  (45) 

Efficiency: 

n  =  po,  (46) 

Apparent  Efficiency: 

Pi  =  £,  (47) 

etc. 

216.  While  excessive  values  of  the  short-circuit  current  under 
the  commutator  brushes,  74,  give  bad  commutation,  due  to  ex- 
cessive current  densities  under  the  brushes,  the  best  commuta- 
tion corresponds  not  to  the  minimum  value  of  I\ — as  the  zero 
value  at  synchronism  in  the  repulsion  motor — but  to  that  value 
of  74  for  which  the  sudden  change  of  current  in  the  armature 
coil  is  a  minimum,  at  the  moment  where  the  coil  leaves  the  com- 
mutator brush. 

/4  is  the  short-current  in  the  armature  coil  during  commuta- 
tion, reduced  to  the  armature  circuit,  /i,  by  the  ratio  of  effective 
turns : 

short-circuited  turns  under  brushes 


_ 


total  effective  armature  turns 


The  actual  current  in  the  short-circuited  coils  during  commuta- 
tion then,  is: 

U  =  ~,  (49) 

C4 

or,  if  we  denote: 

•       ~  =  A4,  (50) 

04 

where  A4  is  a  fairly  large  quantity,  and  substitute  (24),  it  is: 

£2)  (    . 

• 


Before  an  armature  coil  passes  under  the  commutator  brushes, 
it  carries  the  current,  —  /ij  while  under  the  brushes,  it  carries 
the  current,  /%;  and  after  leaving  the  brushes,  it  carries  the  cur- 
rent, -f/i. 


392  ELECTRICAL  APPARATUS 

While  passing  under  the  commutator  brushes,  the  current  in 
the  armature  coils  must  change  from,  —  /i,  to  /'4,  or  by: 

/'„  =  7'4  +  /i.  (52) 

In  the  moment  of  leaving  the  commutator  brushes,  the  cur- 
rent in  the  armature  coils  must  change  from,  I\  to  +  /i,  or  by: 

/,=/!-    /'  4-  (53) 

The  value,  I'g,  or  the  current  change  in  the  armature  coils 
while  entering  commutation,  is  of  less  importance,  since  during 
this  change  the  armature  coils  are  short-circuited  by  the  brushes. 

Of  fundamental  importance  for  the  commutation  is  the  value, 
Ig,  of  the  current  change  in  the  armature  coils  while  leaving  the 
commutator  brushes,  since  this  change  has  to  be  brought  about 
by  the  resistance  of  the  brush  contact  while  the  coil  approaches 
the  edge  of  the  brush,  and  if  considerable,  can  not  be  completed 
thereby,  but  the  current,  /<,,  passes  as  arc  beyond  the  edge  of  the 
brushes. 

Essential  for  good  commutation,  therefore,  is  that  the  current, 
Ig,  should  be  zero  or  a  minimum,  and  the  study  of  the  commu- 
tation of  the  single-phase  commutator  thus  resolves  itself  largely 
into  an  investigation  of  the  commutation  current,  Ig,  or  its  abso- 
lute value,  ig. 

The  ratio  of  the  commutation  current,  ig,  to  the  main  armature 
current,  ii,  can  be  called  the  commutation  constant: 

k  =  $•  (54) 

^l 

For  good  commutation,  this  ratio  should  be  small  or  zero. 

The  product  of  the  commutation  current,  ig,  and  the  speed,  S, 
is  proportional  to  the  voltage  induced  by  the  break  of  this  cur- 
rent, or  the  voltage  which  maintains  the  arc  at  the  edge  of  the 
commutator  brushes,  if  sufficiently  high,  and  may  be  called  the 
commutation  voltage: 

ec  =  Sig.  (55) 

In  the  repulsion  motor,  it  is,  substituting  (23)  and  (51)  in  (53), 
and  dropping  the  term  with  X4,  as  of  secondary  order: 
Commutation  Current: 


(56) 

ZK 


SINGLE-PHASE  COMMUTATOR  MOTORS         393 

Commutation  Constant: 


ij, 
h 


1  + 


1  + 


(57) 


Or,  denoting: 


substituting  (32)  and  expanding: 
T  -  e  i1  ~Co  ^a"  +  (*  ~  ^  «;4l 
7,  e  {1-Co  [Stt77  +  (1  ~  £2 


4  -  Sot*]} 


(58) 


(59) 


and,  absolute: 


/ 

\ 


-c0[Sa"  +  (1  -  S2)a'4]}2  +  Co2  {(1-  S2)  a"*  -  ^a'}2 
_  _  (60) 
{I  -  Co[Sa"  +  (1  -  >S2)-aM}2  +  Co2  {(1  ~  ^S2)  ^4  -  Sa'}2 


(61) 

Perfect  commutation,  or  Ig  =  0,  would  require  from  equation 

(58): 

1  -  Co[Sa"  +  (1->S2)  a'J  =  0, 
(1  -  S2)  «/74-  Sat'  =  0; 
or: 


(62) 


1  -  CQSa" 


Co(l->S2)' 


1  - 


(63) 


This  condition  can  usually  not  be  fulfilled. 
The  commutation  is  best  for  that  speed,  S,  when  the  commu- 
tation current,  iot  is  a  minimum,  that  is  : 


dS 


hence: 


(64) 


394  ELECTRICAL  APPARATUS 

This  gives  a  cubic  equation  in  S}  of  which  one  root,  0  <  Si  <  1, 
represents  a  minimum. 

The  relative  commutation,  that  is,  relative  to  the  current  con- 
sumed by  the  motor,  is  best  for  the  value  of  speed,  $2,  where  the 
commutation  factor,  k,  is  a  minimum,  that  is: 

'  .§-0.  .       H!      (65) 

217.  The  power  output  of  the  repulsion  motor  becomes  zero 
at  the  approximate  speed  given  by  substituting  P  =  0  in  the 
approximate  equation  (40),  as: 

1 


and  above  this  speed,  the  power,  P,  is  negative,  that  is,  the 
repulsion  motor  consumes  power,  acting  as  brake. 

This  value,  $0,  however,  is  considerably  reduced  by  using  the 
complete  equations  (39),  that  is,  considering  the  effect  of  the 
short-circuit  current  under  the  brushes,  etc. 

For  S  <  0,  P  <  0;  that  is,  the  power  is  negative,  and  the 
machine  a  generator,  when  driven  backward,  or,  what  amounts 
to  the  same  electrically,  when  reversing  either  the  field-circuit, 
/o,  or  the  primary  energy  circuit,  /2.  In  this  case,  the  machine 
then  is  a  repulsion  generator. 

The  equations  of  the  repulsion  generator  are  derived  from  those 
of  the  repulsion  motor,  given  heretofore,  by  reversing  the  sign 
of  & 

The  power,  P4,  of  the  short-circuit  current  under  the  brushes 
reverses  at  synchronism,  and  becomes  negative  above  synchron- 
ism. The  explanation  is:  This  short-circuit  current,  /4,  and  a 
corresponding  component  of  the  main  current,  /i,  are  two  cur- 
rents produced  in  quadrature  in  an  armature  or  secondary,  short- 
circuited  in  two  directions  at  right  angles  with  each  other,  and 
so  offering  a  short-circuited  secondary  to  the  single-phase  pri- 
mary, in  any  direction,  that  is,  constituting  a  single-phase  in- 
duction motor.  The  short-circuit  current  under  the  brushes  so 
superimposes  in  the  repulsion  motor,  upon  the  repulsion-motor 
torque,  a  single-phase  induction-motor  torque,  which  is  positive 
below  synchronism,  zero  at  synchronism,  and  negative  above 
synchronism,  as  induction-generator  torque.  It  thereby  lowers 


SINGLE-PHASE  COMMUTATOR  MOTORS        395 

the  speed,  So,  at  which  the  total  torque  vanishes,  and  reduces 
the  power-factor  and  efficiency. 

218.  As  an  example  are  shown  in  Fig.  188  the  characteristic 
curves  of  a  repulsion  motor,  with  the  speed,  S}  as  abscissae,  for 
the  constants: 

Impressed  .voltage:  e     =  500  volts. 
Exciting  impedance,  main  field:  Z    =  0.25    +  3  j  ohms, 
cross  field:  Zf  =  0.25    +  2.5  j  ohms. 


700 


500 


100 


e  -  500  VOLTS 

Z=  0.25+  3  j  Zi=0.025  +  0.075  j 

2=0.25  +  2.53  2 2=  0.025  +  0.075 j 

Z0=0.1  +  0.3J  Z4=7.5  +  10 j 
C  o=0.4  04=0.04 


0        02       04       06       0.8       1.0       1.2       1.4       1.6       1.8       2.0      2.2       2.4 


I      I      I      I      I      II      I     I      I 


700 


450 


350u 

300^ 

250 

200 

150 

100 


FIG.  188. 


Self-inductive  impedance,  main  field :  Z0 

cross  field:  Z2 
armature:  Z\ 
brush  short-circuit:  Z4 
Reduction  factor,  main  field:  c0 
brush  short-circuit:  c4 
Hence: 

Z3  =  0.08  +  0.60  j  ohms. 
A   =  0.835  -  0.014  j. 

j   =  a'  +  ja"  =  1. 20  + 0.02  j. 

Xi  =  0.031  -  0.007  j. 
X4  '=  0.179  +  0.087  j. 
A,  =  4.475  + 2.175  j. 
A*  =  0.202  -  0.010  j. 


=  0.1      +  0.3  j  ohms. 
=  0.025  +  0.075  j  ohms. 
=  0.025  +  0.075  j  ohms. 
=  7.5      -f  10  j  ohms. 
=  0.4. 
=  0.04. 


396  ELECTRICAL  APPARATUS 

Then,  substituting  in  the  preceding  equations  : 

K  =  (0.204  -  0.035  S)  -  j  (0.031  +  0.328  S), 
ZK  =  (0.144  +  0.975  S)  +  j  (0.604  -  0.187  S). 

Primary  or  Supply  Current  : 

,     _  500  {(1.031  -  0.179  S2)  -j  (0.007  +  0.087  S2)  } 

ZK 

Secondary  or  Armature  Current: 

j        500  {  (  1  +  0.048  S  -  0.179  S2)  +  j  0.4  S  -  0.087  S2)} 

ZK 
Brush  Short-circuit  Current: 

and  absolute: 


j    _  500  (1  -  S2)  (0.072  -  0.035  j) 
K~ 


40  (1  - 


m 
Commutation  Factor: 

k  =      K1-508"^2  ~  Q-673)2  +  (0.718  -  0.4  8  -  0.704  £2j" 
\  (0.697  +  0.4  S  -  0.014  )2 

Main  E.m.f.  of  Rotation: 

v,        500  S  (4.052  +  0.792  7) 
El=  ~ZK~ 

Commutation  E.m.f.  of  Rotation: 

,        500  £2  (0.4-  4.8  j) 

E  4  =       ~iir~ 

Power  of  Main  Armature  Circuit: 

2^0  iSf 
pl  =  ^^  (4.052  -  0.122  S  -  0.657  S2),  in  kw. 

Power  of  Brush  Short-circuit: 

49.2  S2  (1  -  /S2)   . 
Pt=  -^-      ^,mkw. 

Total  Power  Output: 
p  =  250^  ( 

tYl 

Torque  : 


D  =  ^  (4.052  +  0.075  S  -  0.657  S2  -  0.197  S3), 


etc. 


SINGLE-PHASE  COMMUTATOR  MOTORS        397 

These  curves  are  derived  by  calculating  numerical  values  in 
tabular  form,  for  S  =  0,  0.2,  0.4,  0.6,  0.8,  1.0,  1.2,  1.4,  1.6,  1.8, 
2.0,  2.2,  2.4. 

As  seen  from  Fig.  188,  the  power-factor,  p,  rises  rapidly,  reach- 
ing fairly  high  values  at  comparatively  low  speeds,  and  remains 
near  its  maximum  of  90  per  cent,  over  a  wide  range  of  speed. 
The  efficiency,  77,  follows  a  similar  curve,  with  90  per  cent,  maxi- 
mum near  synchronism.  The  power,  P,  reaches  a  maximum  of 
192  kw.  at  60  per  cent,  of  synchronism — 450  revolutions  with  a 
four-pole  25-cycle  motor — is  143  kw.  at  synchronism,  and  van- 
ishes, together  with  the  torque,  D,  at  double  synchronism.  The 
torque  at  synchronism  corresponds  to  143  kw.,  the  starting 
torque  to  657  synchronous  kw. 

The  commutation  factor,  k,  starts  with  1.18  at  standstill,  the 
same  value  which  the  same  motor  would  have  as  series  motor, 
but  rapidly  decreases,  and  reaches  a  minimum  of  0.23  at  70  per 
cent,  of  synchronism,  and  then  rises  again  to  1.00  at  synchron- 
ism, and  very  high  values  above  synchronism.  That  is,  the 
commutation  of  the  repulsion  is  fair  already  at  very  low  speeds, 
becomes  very  good  somewhat  below  synchronism,  but  poor  at 
speeds  considerably  above  synchronism:  this  agrees  with  the  ex- 
perience on  such  motors. 

In  the  study  of  the  commutation,  the  short-circuit  current 
under  the  commutator  brushes  has  been  assumed  as  secondary 
alternating  current.  This  is  completely  the  case  only  at  stand- 
still, but  at  speed,  due  to  the  limited  duration  of  the  short-circuit 
current  in  each  armature  coil — the  time  of  passage  of  the  coil 
under  the  brush — an  exponential  term  superimposes  upon  the 
alternating,  and  so  modifies  the  short-circuit  current  and  thereby 
the  commutation  factor,  the  more,  the  higher  the  speed,  and 
greater  thereby  the  exponential  term  is.  The  determination  of 
this  exponential  term  is  beyond  the  scope  of  the  present  work, 
but  requires  the  methods  of  evaluation  of  transient  or  momentary 
electric  phenomena,  as  discussed  in  "Theory  and  Calculation  of 
Transient  Electric  Phenomena  and  Oscillations." 

B.     Series  Repulsion  Motor 

219.  As  further  illustration  of  the  application  of  these  funda- 
mental equations  of  the  single-phase  commutator  motor,  (1)  to 
(6),  a  motor  may  be  investigated,  in  which  the  four  independent 
constants  are  chosen  as  follows: 


398  ELECTRICAL  APPARATUS 

1.  Armature  and  field  connected  in  series  with  each  other. 
That  is: 

El  +  c0#o  =  E  =  61,  (67) 

where : 

c0  =  reduction  factor  of  field  winding  to  armature;  that  is, 

field  turns 
ratio  of  effective  armature  turns' 

It  follows  herefrom: 

/o  =  c0/i.  (68) 

2.  The  e.m.f.  impressed  upon  the  compensating  winding  is 
given,  and  is  in  phase  with  the  e.m.f.,  ei,  which  is  impressed  upon 
field  plus  armature: 

E2  =  e2.  (69) 

That  is,  E2  is  supplied  by  the  same  transformer  or  compensator 
as  61,  in  series  or  in  shunt  therewith. 

3.  No  rotor-exciting  circuit  is  used: 

/*  =  0,  (70) 

and  therefore: 

4.  No  rotor-exciting  brushes,  or  brushes  in  quadrature  posi- 
tion with  the  main-armature  brushes,  are  used,  and  so: 

/•  =  0,  (71) 

that  is,  the  armature  carries  only  one  set  of  brushes,  which  give 
the  short-circuit  current,  /4. 

Since  the  compensating  circuit,  e2,  is  an  independent  circuit, 
it  can  be  assumed  as  of  the  same  number  of  effective  turns  as 
the  armature,  that  is,  e2  is  the  e.m.f.  impressed  upon  the  com- 
pensating circuit,  reduced  to  the  armature  circuit.  (The  actual 
e.m.f.  impressed  upon  the  compensating  circuit  thus  would  be: 

compensating  turns  \ 
c262,  where  c2  =  ratio  effective      armature  turns     ') 

220.  Substituting  (68)  into  (1),  (2),  (3),  and  (5),  and  (1)  and 
(2)  into  (67),  gives  the  three  motor  equations: 

61    =    Z!/!   +   Z'  (/l    -    72)    -  JSZ  (Co/1    -    /4)  (72) 

62  =  Z2/2  +  Z'  (h  -  /i),  (73) 

0  =  Z4/4  +  Z  (/4  -  co/i)  -  JSZ'  (/i  -  /2).          (74) 


SINGLE-PHASE  COMMUTATOR  MOTORS         399 


Substituting  now: 

Yf  =  -yj   =  quadrature,  or  transformer  exciting 
admittance, 

I*  =  x2  =  X'2  -  JX"2, 
r  =  X4  =  X'4  +  JX"4, 


Z 


zf 

7    =  A  =  a   —  ja"  =  impedance   ratio   of   the 

two  quadrature  fluxes, 
Zi  +  Co2  (Z0  +  Z)  =  Zs, 

Z3        A  , 

yr  •-  AS  —  a  3  -f-  Jo:   3, 


and: 


e  =  ei  +  62, 


Adding  (72)  and  (73),  and  rearranging,  gives: 

e  =  Z2/2  +  7i  (Z8  -  jScoZ)  -  /4Z  (c0  -  jS); 


or: 


i=V 

From  (73)  follows: 


(A 


-  /4  (c0  -  jS). 


or: 


and: 


e2r  =  /2  (1  +  X2)  -  7] 

fl    -    /2  (1+   X2)    - 

h  =  /i  (1  -  X2)  - 


From  (74)  follows: 

0  =  /4  (Z  +  Z4)  - 


(75) 


(76) 


(77) 


(78) 


(79) 


Since  74  is  a  small  current,  small  terms,  as  X2,  can  be  neglected 
in  its  evaluation.  That  is:  when  substituting  (78)  in  (79),  X2 
can  be  dropped: 


or: 


approximately. 


(80) 


400  ELECTRICAL  APPARATUS 

Hence,  (80)  substituted  in  (79)  gives  : 


or: 


Hence: 


0  =  74  (Z  +  Z4)  - 


n         4 

0  =  -  -- 

A4 


4- 

H 


74  =  X«  (<*/!-  ^-j. 
and  actual  value  of  short-circuit  current: 


r/4  =  6X4 


jSet.} 


where : 


Co 


6  =  - ,  a  fairly  large  quantity,  and 


c4  =  reduction  factor  of  brush  short-circuit 
to  armature  circuit. 

The  commutation  current  then  is : 


=  /i(l  - 


Substituting  (81)  and  (80)  into  (77),  gives: 


/i 


1  -  jSt\t  (CQ  -  jS)  -  t\2 


or,  denoting: 
it  is: 


Z  AS  —  jSco  —  X4Co  (CQ  —  jS) 

K  =  At  —  jScQ  —  X4c0  (c0  —  jS)  +  X2A, 
co  -  JS)  - 


It  is,  approximately: 


hence : 


^3    =    f    =    Co2, 

X2  =  0, 

K  =  c0  (I  -  c0X4)(c0  -jS), 
T         e  { 1  —  jSt\4  (c0  —  jS) } 


CoZ(l  -  c0X4)  (c0  -  jS) 

^ / L. 

c0Z  (1  —  c0X4)  Ico  —  jS 


(81) 


(82) 


(83) 


(84) 


(85) 

(86) 
(87) 

(88) 


SINGLE-PHASE  COMMUTATOR  MOTORS        401 


Substituting  now  (85)  respectively  (87),  (88)  into  (78),  (81)  (84), 
and  into: 

E\  =  JSZ  (co/i  -  74), 

E\  =  JSZ'  (I,  -  72), 
gives  the 

Equations  of  the  Series  Repulsion  Motor: 


K  =  As  —  jSc0  —  X4c0  (c0  —  jS)  +  X2^L, 
approximately : 

K  =  c0(l  -  c0X4)  (Co  -  jS). 

Inducing,  or  Compensator  Current: 
_  e{l  -jSt\*(c0  -JS)  -  (1  +QX2}        et  (I  -  X2) 

ZK  ~w~r 

approximately: 

te 


CoZ  (1  —  CoX4)  (CQ  —  jS)       CQZ  (1  —  CoX4) 
Armature,  or  Secondary  Current: 

/,  =  e!1~-?<S'X42^-- 
approximately : 


•  *7  / ~\  \\  *i 

LQ£J  \L    —    CoA4>;    I  Co    —  JA 

Brush  Short-circuit  Current: 
eX4  f        1 


-  c0X4)  Ico  -jS 


-  JSt 


X4  -  c0X4) 


approximately  : 


eX4 


Z  (1  —  foX4)  ico  —  ji 
Commutation  Current: 


+  6\4  (1  -  Co)] 


approximately: 


(89) 


(90) 


(91) 


(92) 


(93) 


(94) 


26 


402 


ELECTRICAL  APPARATUS 


Main  E.m.f.  of  Rotation: 


^  3Se        [1  ~X4 

1  —  CoX4  I  CQ  —  .7$ 

approximately : 

,  JSe  (1  -  X4) 

"  (l-CoX4)(t0-jS)' 

Quadrature  E.m.f.  of  Rotation: 

E\  =  +  jSte. 


-  C0) 


(95) 


(96) 


Power  Output: 


Power  Input: 


P  =  Pi  + 


"4,  7J1. 


(98) 


Volt-ampere  Input: 

~D  *         I 

=  e{(l  - 

where  the  small  letters,  ii  and  i'2,  denote  the  absolute  values  of 
the  currents,  /i  and  /2. 

When  i\  and  ^2  are  derived  from  the  same  compensator  or 
transformer  (or  are  in  shunt  with  each  other,  as  branches  of  the 
same  circuit,  if  e\  =  e2),  as  usually  the  case,  in  the  primary  cir- 
cuit the  current  corresponds  not  to  the  sum,  {(1  —  t)  ii  +  ti2}  of 
the  secondary  currents,  but  to  their  resultant,  [(1  —  0  /i  +  </2]1, 
and  if  the  currents,  I\  and  72,  are  out  of  phase  with  each  other, 
as  is  more  or  less  the  case,  the  absolute  value  of  their  resultant 
is  less  than  the  sum  of  the  absolute  values  of  the  components. 
The  volt-ampere  input,  reduced  to  the  primary  source  of  power, 
•then  is : 

and: 

p     <  p 

•*•     OQ      ^     -*•     a* 

p 

From  these  equations  then  follows  the  torque:  D  =  -«-,  the 

power-factor,  p  =  ~ -,  etc. 

These  equations  (90)  to  (99)  contain  two  terms,  one  with,  and 

one  without  t  =  — ,  and  so,  for  the  purpose  of  investigating  the 

e 


SINGLE-PHASE  COMMUTATOR  MOTORS        403 

effect  of  the  distribution  of  voltage,  e,  between  the  circuits,  e\ 
and  62,  they  can  be  arranged  in  the  form  :  F  —  K\  -f  tK2. 

For: 

*  =  0, 

that  is,  all  the  voltage  impressed  upon  the  armature  circuit,  and 
the  compensating  circuit  short-circuited,  these  equations  are 
those  of  the  inductively  compensated  series  motor. 

For: 

t  =  1, 

that  is,  all  the  voltage  impressed  upon  the  compensating  or  in- 
ducing circuit,  and  the  armature  circuit  closed  in  short-circuit, 
that  is,  the  armature  energizing  the  field,  the  equations  are  those 
of  the  repulsion  motor  with  secondary  excitation. 

For: 

t>  1, 

a  reverse  voltage  is  impressed  upon  the  armature  circuit. 
Study  of  Commutation 

221.  The  commutation  of  the  alternating-current  commutator 
motor  mainly  depends  upon: 

(a)  The  short-circuit  current  under  the  commutator  brush, 

which  has  the  actual  value:  I\  =  —  •     High  short-circuit  current 

c\ 

causes  arcing  under  the  brushes,  and  glowing,  by  high  current 
density  : 

(b)  The  commutation  current,  that  is,  the  current  change  in 
the  armature  coil  in  the  moment  of  leaving  the  brush  short-cir- 
cuit, I6  =  Ii  —  I'i.     This  current,  and  the  e.m.f.  produced  by 
it,  SIg,  produce  sparking  at  the  edge  of  the  commutator  brushes, 
and  is  destructive,  if  considerable. 

(a)  Short-circuit  Current  under  Brushes 

Using  the  approximate  equation  (93),  the  actual  value  of  the 
short-circuit  current  under  the  brushes  is: 


•    (100) 

where  : 

b  =  —  ,  or  T  =  reduction  factor  of  short-circuit  under  brushes, 

04  0 


404  ELECTRICAL  APPARATUS 

to  field  circuit,  that  is: 

,   =  number  of  field  turns  , 

number  of  effective  short-circuit  turns' 

hence  a  large  quantity. 

The  absolute  value  of  the  short-circuit  current,  therefore,  is: 


c0z  [1  —  c0X4]  (c02  -f-  AS2) 
hence  a  minimum  for  that  value  of  t,  where : 

/  =  Co2  +  AS2  (1  -  t  (c02  +  AS2))2  =  minimum,  or 
=  1  —  t  (Co2  +  AS2)  =  0,  hence, 

i  =  3?T^' 

and: 

.     ~  •    -.8  -      IT 

That  is,  t  =  —  =  -~ —    — o  gives   minimum  short-circuit   cur- 

e       02  T  Co 

rent   at  speed,  AS,  and  inversely,   speed    AS  =  */-  —  c02,  gives 

minimum  short-circuit  current  at  voltage  ratio,  t. 

For  t  =  1,  or  the  repulsion  motor  with  secondary  excitation, 
the  short-circuit  current  is  minimum  at  speed,  AS  =  \/l  —  c02,  or 

somewhat  below  synchronism,  and  is  i\  —  ~~,  while  in  the  re- 
pulsion motor  with  primary  excitation,  the  short-circuit  current 
is  a  minimum,  and  equals  zero,  at  synchronism  AS  =  1. 

The  lower  the  voltage  ratio,  t  =  — ,  the  higher  is  the  speed,  AS, 

€ 

at  which  the  short-circuit  current  reaches  a  minimum. 

The  short-circuit  current,  /'4,  however,  is  of  far  less  importance 
than  the  commutation  current,  Ig. 

(b)  Commutation  Current 

222.  While  the  value,  J'g  =  I\  +  Ii,  or  the  current  change  in 
the  armature  coils  while  entering  commutation,  is  of  minor  im- 
portance, of  foremost  importance  for  good  commutation  is  that 
the  current  change  in  the  armature  coils,  when  leaving  the  short- 
circuit  under  the  brushes: 

/,  =  /i-  l\  (103) 

is  zero  or  a  minimum. 


SINGLE-PHASE  COMMUTATOR  MOTORS        405 


Using  the  approximate  equation  of  the  commutation  current 
(94),  it  is: 


—  c0 


c0Z(l  -  c0X4)(c0  - 
and,  denoting: 


-^  { 1  -  X4&  +  JS  (c0  -  JS)  t\J>  ) ;    (104) 


it  is,  expanded: 


hence,  absolute: 


X4    =    X'4   4-  JX"4, 


-  j  [X"46  -  ^6  (coX' 


;     (105) 


S2 


-  X'46 


-       "2 


c0X"4)] 


+  SX"4)p, 
(106) 
where  [1  —  c0X4]  denotes  the  absolute  value  of  (1  —  CoX4). 

The  commutation  current  is  zero,  if  either  S  =  °°  ,  that  is, 
infinite  speed,  which  is  obvious  but  of  no  practical  interest,  or 
the  parenthesis  in  (105)  vanishes. 

Since  this  parenthesis  is  complex,  it  vanishes  when  both  of 
its  terms  vanish.  This  gives  the  two  equations: 

1  -  V46  +  Sib  OSX'4  -  c0X"4)  =  0, 
X"46  -  Stb  (c0X'4  +  SX"4)  =  0. 


(107) 


From  these  two  equations  are  calculated  the  two  values,  the 
speed,  S,  and  the  voltage  ratio,  t,  as: 


Q        c0  (bX42  -  X'4) 

00    =  ,  „  - 

X     4 
\".2 

to  = 


hence  : 


X// 
4 


(108) 


For  instance,  if: 


Z   =0.25  +  3j, 
Z,  =  5       +  2.5  j: 


406  ELECTRICAL  APPARATUS 

hence : 

X4  =  z  ^  z^  =  0.307  +  0.248  j  =  X'4  +  JX"4, 

c0  =  0.4, 

c4  =  0.04; 
hence: 

b  =  10; 
and  herefrom: 

So  =  2.02, 

*o  =  0.197 

that  is,  at  about  double  synchronism,  for  ez  =  te  =  0.197e,  or 
about  20  per  cent,  of  e,  the  commutation  current  vanishes. 

In  general,  there  is  thus  in  the  series  repulsion  motor  only  one 
speed,  So,  at  which,  if  the  voltage  ratio  has  the  proper  value,  to, 
the  commutation  current,  ig,  vanishes,  and  the  commutation  is 
perfect.  At  any  other  speed  some  commutation  current  is  left, 
regardless  of  the  value  of  the  voltage  ratio,  t. 

With  the  two  voltages,  e\  and  e2,  in  phase  with  each  other,  the 
commutation  current  can  not  be  made  to  vanish  at  any  desired 
speed,  S. 

223.  It  remains  to  be  seen,  therefore,  whether  by  a  phase  dis- 
placement between  e\  and  e2,  that  is,  if  e%  is  chosen  out  of  phase 
with  the  total  voltage,  e,  the  commutation  current  can  be  made 
to  vanish  at  any  speed,  S,  by  properly  choosing  the  value  of  the 
voltage  ratio,  and  the  phase  difference. 

Assuming,  then,  e2  out  of  phase  with  the  total  voltage,  e,  hence 
denoting  it  by : 

Ez  =  ez  (cos  02  -  j  sin  02),  (109) 

the  voltage  ratio,  t}  now  also  is  a  complex  quantity,  and  expressed 
by: 

T  =  ~  =  t  (cos  02  -  j  sin  02)  =  t'  -  ji".  (110) 

Substituting  (110)  in  (105),  and  rearranging,  gives: 

j 7 — r~7 r^r  { [1   —  X  46  -{-  o£  b  (oX  4  —  CoX    4) 

L  —  CoX4)  (CQ  —  J£>) 

+  St"b  (c0X'4  +  SX"4)]  -  j[X"46  -  St'b  (c0X'4  +  <?X"4) 

and  this  expression  vanishes,  if: 

1  -  V46  +  St'b  (S\\  -  c0X"4)  +  £r&  (c0X'4  +  £X"4)  =  0,  ' 
X"46  -  Stfb  (c0X'4  +  S\'\)  +  St"b  (SX'4  -  c0X"4)  =0: 


SINGLE-PHASE  COMMUTATOR  MOTORS        407 

and  herefrom  follows: 
,    =  SW  -  SX'4  +  CoX" 


£&X42(S2  +  co2)  Co2  +  /S2  I 

42  -  c0X'4-£X"4  1        [Co     C0X'4 


\  co2  +  S2  i  S 

or  approximately: 


(112) 


C°2  +C0S  <113) 

\"  —          c° 


t"  =  0  substituted  in  equation  (112)  gives  S  =  So,  the  value 
recorded  in  equation  (108). 

It  follows  herefrom,  that  with  increasing  speed,  S,  t'  and  still 
more  t",  decrease  rapidly.  For  S  =  0,  t'  and  t"  become  infinite. 
That  is,  at  standstill,  it  is  not  possible  by  this  method  to  produce 
zero  commutation  current. 

The  phase  angle,  02,  of  the  voltage  ratio,  T  =  t'  —  jt",  is  given 
by: 


tan  02  =  T 

CoOA4"   —   CoA  4   —   OA     4. 

(114) 

')'"    $6X42    —   >SX'4  +    CoX'Y 

rearranged,  this  gives: 

c0  sin  02 

+  8  cos  02        6X42  -  X'4  . 

(115) 

Co  sin  02 

-S  sin  02             X"4 

and,  denoting: 

8 

—  =  tan  <r, 

Co 

(116) 

where  cr  may  be  called  the  "speed  angle,"  it  is,  substituted  in 
(115): 

tan  (fc  +  <r)  =  ^X4^4,  (117) 

A     4 

=  constant; 
hence: 

02  +  er  =  T,  (118) 

and: 

02  =  T  -  *•  (119) 

-^-77— —  is  a  large  quantity,  hence  y  near  90°. 

A     4 

<r  is  also  near  90°  for  all  speeds,  S,  except  very  slow  speeds,  since 
in  (116)  Co  is  a  small  quantity. 


408  ELECTRICAL  APPARATUS 

Hence  62  is  near  zero  for  all  except  very  low  speeds. 

For  very  low  speeds,  a  is  small,  and  02  thus  large  and  positive. 

That  is,  the  voltage,  Ez,  impressed  upon  the  compensating 
circuit  to  get  negligible  commutation  current,  must  be  approxi- 
mately in  phase  with  e  for  all  except  low  speeds.  At  low  speeds, 
it  must  lag,  the  more,  the  lower  the  speed.  Its  absolute  value 
is  very  large  at  low  speeds,  but  decreases  rapidly  with  increasing 
speed,  to  very  low  values. 

For  instance,  let,  as  before : 

X4  =  0.304  -  0.248  j, 
Co  =  0.4, 
b  =  10; 
it  is: 

tan  (62  +  o")  =  5.05, 
02  +  ,  =  79°; 
hence: 


62  =  0  for  a  =  79°;  hence,  by  (116),  SQ  =  2.02,  or  double  syn- 
chronism. Above  this  speed,  02  is  leading,  but  very  small,  since 
the  maximum  leading  value,  for  infinite  speed,  S  =  <*> ,  is  given 
by  a  =  90°,  as,  02  =  —11°.  Below  the  speed,  So,  02  is  positive, 
or  lagging; 

for  S  =  1,  it  is  o-  =  68°,  02  =  +11°,  hence  still  approximately 
in  phase; 

for  S  =  0.4,  it  is  (j  =  45°,  02  =  34°;  hence  E2  is  still  nearer  in 
phase  than  in  quadrature  to  e. 

The  corresponding  values  of  T  =  t'  +  t"  are,  from  (112): 

S  =  2.02,  02  =  0,  T  =  0.197,  t  =  0.197, 

5  =  1,        02  =  +11°,      T  =  0.747  +0.140  j,      t  =  0.760, 
S  =  0.4,     02  =  34°,          T  =  3.00    -2.00   j,      t  =  3.61. 

224.  The  introduction  of  a  phase  displacement  between  the 
compensating  voltage,  Ez,  and  the  total  voltage,  e,  in  general  is 
more  complicated,  and  since  for  all  but  the  lowest  speeds  the 
required  phase  displacement,  02,  is  small,  it  is  usually  sufficient 
to  employ  a  compensating  voltage,  ez,  in  phase  with  e. 

In  this  case,  no  value  of  t  exists,  which  makes  the  commutation 
current  vanish  entirely,  except  at  the  speed,  S0. 

The  problem  then  is,  to  determine  for  any  speed,  S,  that  value 


SINGLE-PHASE  COMMUTATOR  MOTORS        409 

of  the  voltage  ratio,  t,  which  makes  the  commutation  current,  ig, 
a  minimum.     This  value  is  given  by: 

f  =  0,  (120) 

where  ig  is  given  by  equation  (106). 

Since  equation  (106)  contains  t  only  under  the  square  root, 
the  minimum  value  of  i6  is  given  also  by: 

^-0 
dt   '    "' 

where  : 

K  =  [l-  b\\  +  Stb  (S\'4  -  c0X"4)]2  +  [6X"4  -  Stb  (c0X'4  +  SX"4)]2. 
Carrying  out  this  differentiation,  and  expanding,  gives: 


_  -  S\\  +  CoX"4  _          1        /         S\\-  c0X"4 

~ 


+  S2)          [c02  +  S2  r  Sb\,2       j 

This  is  the  same  value  as  the  real  component,  Z',  of  the  complex 
voltage  ratio,  TI,  which  caused  the  commutation  current  to 
vanish  entirely,  and  was  given  by  equation  (112). 

It  is,  approximately: 

t  =  ^2  ^_  S2-  (122) 

Substituting  (121)  into  (105)  gives  the  value  of  the  minimum 
commutation  current,  igo. 

Since  the  expression  is  somewhat  complicated,  it  is  preferable 
to  introduce  trigonometric  functions,  that  is,  substitute : 

tan  d  =  ~>  (123) 

A  4 

where  8  is  the  phase  angle  of  X4,  and  therefore  r 

X"4  =  X4sin5,  }  } 

X'4    =  X4  cos  5,  (U4) 


and  also  to  introduce,  as  before,  the  speed  angle  (116) : 

8 

tan  <j  — — > 

Co 


(125) 


Q  =  Vc<>2  + 

hence: 

S  =  q  sin  a, (126) 
Co  =  q  cos  (7. 


410 


ELECTRICAL  APPARATUS 


Substituting  these  trigonometric  values  into  the  expression 
(121)  of  the  voltage  ratio  for  minimum  commutation  current, 
it  is: 

sin  *  ~  S 


Substituting  (117)  into  (106)  and  expanding  gives  a  relatively 
simple  value,  since  most  terms  eliminate: 

Ig  =  e  {[cos2  (o-  —  d)  +  b\  (sin  a  sin  (<r  —  6)  —  cos  5)] 

j  [  sin  (a  —  8)  cos  (a-  —  8)  —  6X4  (sin-q-  cos  (<r  —  d)  —  sin  d)]} 
-  C0X4,  (c0  +  jS) 

(128) 

(129) 

*'°  "      Cos  [1  -  c0X4]  (c02  +  S2)  '  (130) 

From  (129)  and  (130)  follows,  that  ittQ  =  0,  or  the  commutation 
current  vanishes,  if: 

cos  (a  —  d)  —  6X4  cos  o-  =  0,  (131) 

or: 

SX"4  -  c0  (X426  -  X'4)  =  0. 

This  gives,  substituting,  X"4  =  \/X42  —  X'42,  and  expanding: 
X4 


and  the  absolute  value: 

e  ( cos  (<T  —  6)  —  6X4  cos  0-) . 

*'°  =     c0z[l  -  c0X4]  VcTT^"2  ' 
or,  resubstituting  for  o-  and  5 : 

-Co(X42fr  -X/)} 


±  S      ^2  -  Co2  (62X42  -  1)}, 
6X4c0 


(132) 


From  (131)  follows: 

COS  ((7  —  d)    =   6X4  COS  (T. 

Since  cos  (<r  —  6)  must  be  less  than  one,  this  means: 

6X4  cos  a  <  1, 


or: 


X4< 


b  cos 


X4< 


or,  inversely: 


>  Co  \/&2X42  -  1. 


(133) 


SINGLE-PHASE  COMMUTATOR  MOTORS        411 

That  is: 

The  commutation  current,  ig,  can  be  made  to  vanish  at  any 
speed,  S,  at  given  impedance  factor,  \4,  by  choosing  the  phase 
angle  of  the  impedance  of  the  short-circuited  coil,  6,  or  the  resist- 
ance component,  X',  provided  that  \4  is  sufficiently  small,  or  the 
speed,  S,  sufficiently  high,  to  conform  with  equations  (133). 

From  (132)  follows  as  the  minimum  value  of  speed,  S,  at  which 
the  commutation  current  can  be  made  to  vanish,  at  given  \i' 

Si  =  Co  V&2X42  -  1, 
and: 


hence: 


For  high  values  of  speed,  S,  it  is,  approximately: 

cos  (a-  —  5)  =  0, 
a  -  5     =  90°, 

tan  <r  =  — ; 

Co 

hence:  or  =  90° 

5  =  0 

X'4  =  X4. 

That  is,  the  short-circuited  coil  under  the  brush  contains  no 
inductive  reactance,  hence: 

At  low  and  medium  speeds,  some  inductive  reactance  in  the 
short-circuited  coils  is  advantageous,  but  for  high  speeds  it  is 
objectionable  for  good  commutation. 

225.  As  an  example  are  shown,  in  Figs.  189  and  192,  the  char- 
acteristic curves  of  series-repulsion  motors,  for  the  constants: 

Impressed  voltage:  e  =  500  volts, 

Exciting  impedance,  main  field:  Z  =  0.25  +  3j  ohms, 

Exciting  impedance,  cross  field:  Z'=  0.25  +2.5  j  ohms, 
Self-inductive  impedance,  main 

field:  Z0=  0.1  +  0.3  j  ohms, 
Self-inductive  impedance,   cross 

field:  Z2  =  0.025  +  0.075  j  ohms, 


412 


ELECTRICAL  APPARATUS 


0  =  500    VOLTS        -§^-  =  0 
Z=0.25+3j       Z:=  0.025 +  0.075J 
Z«=0.25+2.5j   Z2=  0.025 +  0.075J 
Z0=0.1  +  0.3 j  2  i=      7.5-1-  10j 
=0.4  C  4=  0.04 


700 

650 

COO 

550 

500 

450</> 

400H 


300d 
250* 
200 
150 
100 
50 


0          0.2       0.4       0.6       0.8       1.0       19       14       IK       is       on       09      o<         ° 

FIG.  189. — Inductively  compensated  series  motor. 


6  =  500    VOLTS         f-2  =  0.2 
Z=0.25+3j      Zi=  0.025 +0.075J 
Z'=0.25+2.5j  Z2  =  0.02 5  +0.07 5 j 
Z0=0.1  +0..3J  Z4=    7.5  +  lOj 
C0=0.4  C^  =  0.04 


0.2       0.4       0.6       0.3       1.0       1.2       1.4       1.6 


650 
600 
550 
500 

450  <n 
400  H 

350< 
3000 
2505 
200 
150 
100 
50 


2.0       2.2       2.4 


FIG.  190. — Series  repulsion  motor. 


6  =  500  VOLTS    -§-*  =  0.5 
Z=0.25  +  3j 

Z'=  0.25  +  2. 5 j    Z2=0.025+0.075^ 
Z0=0.1  +0.3J    Z4  =    7.5  +  10J     I   / 
C^  =  0.04 


0.2        0.4       0.6       0.8       1.0        1.2       1.4        1.6        1.8        2.0       2.2       2.4 
SPEED 

FIG.  191. — Series  repulsion  motor. 


3000 
250^ 
200 
150 
100 
50 


SINGLE-PHASE  COMMUTATOR  MOTORS        413 


Self-inductive  impedance  arma- 
ture: Zi=  0.025  +  0.075  j  ohms, 

Self-inductive  impedance,  brush 

short-circuit:  Z4=  7.5  +  10  j  ohms, 

Reduction  factor,  main  field:  c0  =  0.4, 

brush  short-circuit  €4  =  0.04; 

that  is,  the  same  constants  as  used  in  the  repulsion  motor, 
Fig.  188. 

Curves  are  plotted  for  the  voltage  ratios : 

t  =  0:  inductively  compensated  series  motor,  Fig.  189. 
t  =  0.2:  series  repulsion  motor,  high-speed,  Fig.  190. 
t  =  0.5:  series  repulsion  motor,  medium-speed,  Fig.  191. 
t  =  1.0:  repulsion  motor  with  secondary  excitation,  low-speed, 
Fig.  192. 


700 


0.2       0.4       0.6       0.8       1.0       1.2       1.4       1.6       1.8        2.0        2.2       2.4 


6  =  500  VOLTS     (-1-2  =  1  ) 
Z=  0.254-3:?       Zi=0  025+0.075:? 
Z=  0.25+  2.5  j    Z2=  0.025 +  0.075J 
Z0=0.1+0.3:?     Z4=7.5.0+10j 
C0=   0.4  C4=  0.04 


400  £ 
350g 
3003 
250* 


FIG.  192. — Repulsion  motor,  secondary  excitation. 

It  is,  from  above  constants : 

Z3  =  Zl  +  Co2  (Z0  +  Z)  =  0.08    +  0.60  j. 


Z 

^ 

Z' 


c4 


=  0.202  -  0.010  j. 
=  0.835  -  0.014  j. 
=  0.031  -  0.007  j. 
=  0.179  +  0.087  j. 
=  10. 


414  ELECTRICAL  APPARATUS 

Hence,  substituting  into  the  preceding  equations: 
(90)     ZK  =  Z3  -  jSc0Z  -  X4c0Z  (c0  -  jS)  +•  X2Z' 

=  (0.160  +  0.975  S)  +  j  (0.590  -  0.187  S), 

(92)        /!  =  ~  -          {jSX4  (co  -  jS)  +  X2} 


(~  °-031  +  °-035  5  ~  °-179 


-j(-  0.007  +  0.072  S  +  0.087  S2)  }  , 
(91)       1  2  =  1  1  (0.969  +  0.007  j)  +  et  (0.010  -  0.096  j), 

(93) 

_e(0.072+0.035.y)+^{(0.016~0.072jS)-j0.045+0.035<S)} 
/4~  ZK 

etc. 

226.  As  seen,  these  four  curves  are  very  similar  to  each  other 
and  to  those  of  the  repulsion  motor,  with  the  exception  of  the 

a 

commutation  current,  ig,  and  commutation  factor,  k  =  —.• 

i 

The  commutation  factor  of  the  compensated  series  motor, 
that  is,  the  ratio  of  current  change  in  the  armature  coil  while 
leaving  the  brushes,  to  total  armature  current,  is  constant  in  the 
series  motor,  at  all  speeds.  In  the  series  repulsion  motors,  the 
commutation  factor,  k,  starts  with  the  same  value  at  standstill, 
as  the  series  motor,  but  decreases  with  increasing  speed,  thus 
giving  a  superior  commutation  to  that  of  the  series  motor,  reaches 
a  minimum,  and  then  increases  again.  Beyond  the  minimum 
commutation  factor,  the  efficiency,  power-factor,  torque  and  out- 
put of  the  motor  first  slowly,  then  rapidly  decrease,  due  to  the 
rapid  increase  of  the  commutation  losses.  These  higher  values, 
however,  are  of  little  practical  value,  since  the  commutation  is 
bad. 

The  higher  the  voltage  ratio,  t,  that  is,  the  more  voltage  is 
impressed  upon  the  compensating  circuit,  and  the  less  upon  the 
armature  circuit,  the  lower  is  the  speed  at  which  the  commuta- 
tion factor  is  a  minimum,  and  the  commutation  so  good  or  perfect. 
That  is,  with  t  =  1,  or  the  repulsion  motor  with  secondary  ex- 
citation, the  commutation  is  best  at  70  per  cent,  of  synchronism, 
and  gets  poor  above  synchronism.  With  t  =  0.5,  or  a  series 
repulsion  motor  with  half  the  voltage  on  the  compensating,  half 
on  the  armature  circuit,  the  commutation  is  best  just  above  syn- 
chronism, with  the  motor  constants  chosen  in  this  instance,  and 


SINGLE-PHASE  COMMUTATOR  MOTORS        415 

gets  poor  at  speeds  above  150  per  cent,  of  synchronism.  With 
t  =  0.2,  or  only  20  per  cent,  of  the  voltage  on  the  compensating 
circuit,  the  commutation  gets  perfect  at  double  synchronism. 

Best  commutation  thus  is  secured  by  shifting  the  supply  vol- 
tage with  increasing  speed  from  the  compensating  to  the  arma- 
ture circuit. 

t  >  1,  or  a  reverse  voltage,  —e\>  impressed  upon  the  armature 
circuit,  so  still  further  improves  the  commutation  at  very  low 
speeds. 

For  high  values  of  t,  however,  the  power-factor  of  the  motor 
falls  off  somewhat. 

The  impedance  of  the  short-circuited  armature  coils,  chosen 
in  the  preceding  example: 

Z4  =  7.5  +  10  j, 

corresponds  to  fairly  high  resistance  and  inductive  reactance  in 
the  commutator  leads,  as  frequently  used  in  such  motors. 

227.  As  a  further  example  are  shown  in  Fig.  193  and  Fig.  194 
curves  of  a  motor  with  low-resistance  and  low-reactance  com- 
mutator leads,  and  high  number  of  armature  turns,  that  is,  low 
reduction  factor  of  field  to  armature  circuit,  of  the  constants : 

£4  =  4  +  2j; 
hence : 

X4  =  0.373  +  0.267  j, 
and: 

Co  =  0.3, 

c4  =  0.03, 

the  other  .constants  being  the  same  as  before. 

Fig.  193  shows,  with  the  speed  as  abscissae,  the  current,  torque, 
power  output,  power-factor,  efficiency  and  commutation  current, 
ig,  under  such  a  condition  of  operation,  that  at  low  speeds  t  =  1.0, 
that  is,  the  motor  is  a  repulsion  motor  with  secondary  excita- 
tion, and  above  the  speed  at  which,  t  =  1.0  gives  best  commuta- 
tion (90  per  cent,  of  synchronism  in  this  example),  t  is  gradually 
decreased,  so  as  to  maintain  ig  a  minimum,  that  is,  to  maintain 
best  commutation. 

As  seen,  at  10  per  cent,  above  synchronism,  ia  drops  below  i, 
that  is,  the  commutation  of  the  motor  becomes  superior  to  that 
of  a  good  direct-current  motor. 

Fig.  194  then  shows  the  commutation  factors,  k  =  —•>    of  the 


416 


ELECTRICAL  APPARATUS 


I      I      I      I      I 


€  =500  VOLTS 
Z=0.25+3j  OHMS     Zi=  0.025  + 0.075  j  OHMS 
Z'=  0.25  +2.5J OHMS  Z2  =  0.025  + 0.075  JOHMS 
Zo=0.1  +  0.3JOHMS  Z<  =  0-4  +  0.2J  OHMS 
C0  =0.3  C4  =  0.03 


1.0       1.2        1.4 
SPEED 

FIG.  193. 


1.6       1.8       2.0       2.2       2.4 


3500 

3005 

250 

200 

150 

100 

50 

0 


0.25 

Z'=0.25  +  2.5JOHMS 
Z0=0.1      +0.3JOHMS 
Zi=0.025+0.075joHMS 
Z2=0.025+0.075JOHMS 
Z4=4-  2  j  OHMS 
C0=0.3 


0        0.20      0.40       0.60       0.80      1.00       1.20      1.40       1.60       1.80 

SPEED  %  OF  SYNCHRONISM 
FIG.  194. 


2.00      2.20      2.40 


SINGLE-PHASE  COMMUTATOR  MOTORS        417 

different  motors,  all  under  the  assumption  of  the  same  constants : 

Z   =  0.25  +  3j, 

Z'  =  0.25  +  2.5  j, 

Z0  =  0.1  +  0.3J, 

Z2  =  0.025  +  0.075  j, 

Z1  =  0.025  +  0.075  j, 

^4  =  4  +2.7, 
Co  =  0.3, 
c4  =  0.03. 

Curve  I  gives  the  commutation  factor  of  the  motor  as  induct- 
ively compensated  series  motor  (t  =  0),  as  constant,  k  =  3.82, 
that  is,  the  current  change  at  leaving  the  brushes  is  3.82  times 
the  main  current.  Such  condition,  under  continued  operation, 
would  give  destructive  sparking. 

Curve  II  shows  the  series  repulsion  motor,  with  20  per  cent,  of 
the  voltage -on  the  compensating  winding,  t  =  0.2;  and 

Curve  III  with  half  the  voltage  on  the  compensating  winding, 
t  =  0.5. 

Curve  IV  corresponds  to  t  =  1,  or  all  the  voltage  on  the  com- 
pensating winding,  and  the  armature  circuit  closed  upon  itself: 
repulsion  motor  with  secondary  excitation. 

Curve  V  corresponds  to  t  =  2,  or  full  voltage  in  reverse  direction 
impressed  upon  the  armature,  double  voltage  on  the  compen- 
sating winding. 

Curve  VI  gives  the  minimum  commutation  factor,  as  derived 
by  varying  t  with  the  speed,  in  the  manner  discussed  before. 

For  further  comparison  are  given,  for  the  same  motor 
constants: 

Curve  VII,  the  plain  repulsion  motor,  showing  its  good  com- 
mutation below  synchronism,  and  poor  commutation  above 
synchronism;  and 

Curve  VIII,  an  overcompensated  series  motor,  that  is,  con- 
ductively  compensated  series  motor,  in  which  the  compensating 
winding  contains  20  per  cent,  more  ampere-turns  than  the  arma- 
ture, so  giving  20  per  cent,  overcompensation. 

As  seen,  overcompensation  does  not  appreciably  improve 
commutation  at  low  speeds,  and  spoils  it  at  higher  speeds. 

Fig.  194  also  gives  the  two  components  of  the  compensating 
e.m.f.,  E2,  which  are  required  to  give  perfect  commutation,  or 
zero  commutation  current: 

27 


418  ELECTRICAL  APPARATUS 

t'v  =  - 2  =  component  in  phase  with  e,  giving  quadrature 

6 

flux: 

z"  i 
t"o  =  -  -  =  component  in  quadrature  with  e,  giving  flux  in 

c/ 

phase  with  e. 

228.  In  direct-current  motors,  overcompensation  greatly 
improves  commutation,  and  so  is  used  in  the  form  of  a  com- 
pensating winding,  commutating  pole  or  interpole.  In  such 
direct-current  motors,  the  reverse  field  of  the  interpole  produces 
a  current  in  the  short-circuited  armature  coil,  by  its  rotation,  in 
the  sanle  direction  as  the  armature  current  in  the  coil  after 
leaving  the  brushes,  and  by  proper  proportioning  of  the  com- 
mutating field,  the  commutation  current,  igj  thus  can  be  made 
to  vanish,  that  is,  perfect  commutation  produced. 

In  alternating-current  motors,  to  make  the  commutation 
current  vanish  and  so  produce  perfect  commutation;  the  current 
in  the  short-circuited  coil  must  not  only  be  equal  to  the  arma- 
ture current  in  intensity,  but  also  in  phase,  that  is,  the  commu- 
tating field  must  not  only  have  the  proper  intensity,  but  also 
the  proper  phase. 

In  paragraph  223  we  have  seen  that  the  commutating  field 
has  the  proper  phase  to  make  ig  vanish,  if  produced  by  a  voltage 
impressed  upon  the  compensating  winding: 

E,  =  Te, 

which  for  all  except  very  low  speeds  is  very  nearly  in  phase  with 
e.  The  magnetic  flux  produced  by  this  voltage,  or  the  corn- 
mutating  flux,  so  is  nearly  in  quadrature  with  e,  and  therefore 
approximately  in  quadrature  with  the  current  in  the  motor, 
at  such  speeds  where  the  current,  i,  is  nearly  in  phase  with  e. 
The  commutating  flux  produced  by  conductive  overcompensa- 
tion, however,  is  in  phase  with  the  current,  i,  hence  is  of  a 
wrong  phase  properly  to  commutate. 

That  is,  in  the  alternating-current  commutator  motor,  the 
commutating  flux  should  be  approximately  in  quadrature  with 
the  main  flux  or  main  current,  and  so  can  not  be  produced  by 
the  main  current  by  overcompensation,  but  is  produced  by  the 
combined  magnetizing  action  of  the  main  current  and  a  sec- 
ondary current  produced  thereby,  since  in  a  transformer  the  re- 
sultant flux  lags  approximately  90°  behind  the  primary  current. 


SINGLE-PHASE  COMMUTATOR  MOTORS         419 

The  same  results  we  can  get  directly  by  investigating  the  com- 
mutation current  of  the  overcompensated  series  motor.  This 
motor  is  characterized  by: 

1.        e  =  Ei  +  c0E0  +  c2E2; 

where   c2  =  1  +  q  =  reduction  factor  of  compensating  circuit 
to  armature. 

2.  /0    -    Co/,   /2    -    C27,   /!    =    /. 

Substituting  into  the  fundamental  equations  of  the  single- 
phase  commutator  motor  gives  the  results: 


'*w 

X4  (c0  -  .?SaA) 
e, 


where  : 

ZK  =  (Zt  +  Z5  +  jSc0Z)  +  jS\,  (c0Z  -  jSqZ') 
To  make  Jg  vanish,  it  must  therefore  be: 


or  approximately: 

.Co    X 

q=-^x' 
or,  with  the  numerical  values  of  the  preceding  instance: 

0.046  -  0.295  j 

q=  ~S~ 

That  is,  the  overcompensating  component,  q,  must  be  approxi- 
mately in  quadrature  with  the  current,  /,  hence  can  not  be  pro- 
duced by  this  current  under  the  conditions  considered  here;  and 
over  compensation,  while  it  may  under  certain  conditions  improve 
the  commutation,  can  as  a  rule  not  give  perfect  commutation 
in  a  series  alternating-current  motor. 

229.  The  preceding  study  of  commutation  is  based  on  the 
assumption  of  the  short-circuit  current  under  the  brush  as 
alternating  current.  This,  however,  is  strictly  the  case  only  at 
standstill,  as  already  discussed  in  the  paragraphs  on  the  repul- 
sion motor.  At  speed,  an  exponential  term,  due  to  the  abrupt 


420  ELECTRICAL  APPARATUS 

change  of  current  in  the  armature  coil  when  passing  under  the 
brush,  superimposes  upon  the  e.m.f.  generated  in  the  short- 
circuited  coil,  and  so  on  the  short-circuit  current  under  the 
brush,  and  modifies  it  the  more,  the  higher  the  speed,  that  is, 
the  quicker  the  current  change.  This  exponential  term  of  e.m.f. 
generated  in  the  armature  coil  short-circuited  by  the  commutator 
brush,  is  the  so-called  " e.m.f.  of  self-induction  of  commutation." 
It  exists  in  direct-current  motors  as  well  as  in  alternating-current 
motors,  and  is  controlled  by  overcompensation,  that  is,  by  a 
commutating  field  in  phase  with  the  main  field,  and  approxi- 
mately proportional  to  the  armature  current. 

The  investigation  of  the  exponential  term  of  generated  e.m.f. 
and  of  short-circuit  current,  the  change  of  the  commutation 
current  and  commutation  factor  brought  about  thereby  and 
the  study  of  the  commutating  field  required  to  control  this 
exponential  term  leads  into  the  theory  of  transient  phenomena, 
that  is,  phenomena  temporarily  occurring  during  and  immedi- 
ately after  a  change  of  circuit  condition.1 

The  general  conclusions  are: 

The  control  of  the  e.m.f.  of  self-induction  of  commutation  of 
the  single-phase  commutator  motor  requires  a  commutating 
field,  that  is,  a  field  in  quadrature  position  in  space  to  the  main 
field,  approximately  proportional  to  the  armature  current  and 
in  phase  with  the  armature  current,  hence  approximately  in 
phase  with  the  main  field. 

Since  the  commutating  field  required  to  control,  in  the  arma- 
ture coil  under  the  commutator  brush,  the  e.m.f.  of  alternation 
of  the  main  field,  is  approximately  in  quadrature  behind  the 
main  field — and  usually  larger  than  the  field  controlling  the 
e.m.f.  of  self-induction  of  commutation — it  follows  that  the 
total  commutating  field,  or  the  quadrature  flux  required  to  give 
best  commutation,  must  be  ahead  of  the  values  derived  in 
paragraphs  221  to  224. 

As  the  field  required  by  the  e.m.f.  of  alternation  in  the  short- 
circuited  coil  was  found  to  lag  for  speeds  below  the  speed  of  best 
commutation,  and  to  lead  above  this  speed,  from  the  position 
in  quadrature  behind  the  main  field,  the  total  commutating 
field  must  lead  this  field  controlling  the  e.m.f.  of  alternation, 
and  it  follows : 

1  See  "Theory  and  Calculations  of  Transient  Electric  Phenomena  and 
Oscillations,"  Sections  I  and  II. 


SINGLE-PHASE  COMMUTATOR  MOTORS         421 

Choosing  the  e.m.f.,  E2,  impressed  upon  the  compensating 
winding  in  phase  with,  and  its  magnetic  flux,  therefore  in  quad- 
rature (approximately),  behind  the  main  field,  gives  a  com- 
mutation in  the  repulsion  and  the  series  repulsion  motor  which 
is  better  than  that  calculated  from  paragraphs  221  to  224,  for  all 
speeds  up  to  the  speed  of  best  commutation,  but  becomes  in- 
ferior for  speeds  above  this.  Hence  the  commutation  of  the 
repulsion  motor  and  of  the  series  repulsion  motor,  when  con- 
sidering the  self-induction  of  commutation,  is  superior  to  the 
calculated  values  below,  inferior  above  the  critical  speed,  that 
is,  the  speed  of  minimum  commutation  current.  The  com- 
mutation of  the  overcompensated  series  motor  is  superior  to  the 
values  calculated  in  the  preceding,  though  not  of  the  same 
magnitude  as  in  the  motors  with  quadrature  commutating  flux. 

It  also  follows  that  an  increase  of  the  inductive  reactance  of  the 
armature  coil  increases  the  exponential  and  decreases  the  alter- 
nating term  of  e.m.f.  and  therewith  the  current  in  the  short- 
circuited  coil,  and  therefore  requires  a  commutating  flux  earlier 
in  phase  than  that  required  by  an  armature  coil  of  lower  reac- 
tance, hence  improves  the  commutation  of  the  series  repulsion 
and  the  repulsion  motor  at  low  speeds,  and  spoils  it  at  high 
speeds,  as  seen  from  the  phase  angles  of  the  commutating  flux 
calculated  in  paragraphs  221  to  224. 

Causing  the  armature  current  to  lag,  by  inserting  external 
inductive  reactance  into  the  armature  circuit,  has  the  same 
effect  as  leading  commutating  flux:  it  improves  commutation  at 
low,  impairs  it  at  high  speeds.  In  consequence  hereof  the  com- 
mutation of  the  repulsion  motor  with  secondary  excitation — 
in  which  the  inductive  reactance  of  the  main  field  circuit  is  in 
the  armature  circuit — is  usually  superior,  at  moderate  speeds, 
to  that  of  the  repulsion  motor  with  primary  excitation,  except 
at  very  low  speeds,  where  the  angle  of  lag  of  the  armature  cur- 
rent is  very  large. 


CHAPTER  XXI 
REGULATING  POLE  CONVERTERS 

230.  With  a  sine  wave  of  alternating  voltage,  and  the  com- 
mutator brushes  set  at  the  magnetic  neutral,  that  is,  at  right 
angles  to  the  resultant  magnetic  flux,  the  direct  voltage  of  a  syn- 
chronous converter  is  constant  at  constant  impressed  alternating 
voltage.     It  equals  the  maximum  value  of  the  alternating  voltage 
between  two  diametrically  opposite  points  of  the  commutator, 
or  "  diametrical  voltage/'  and  the  diametrical  voltage  is  twice 
the  voltage  between  alternating  lead  and  neutral,  or  star  or  Y 
voltage  of  the  polyphase  system. 

A  change  of  the  direct  voltage,  at  constant  impressed  alter- 
nating voltage  (or  inversely),  can  be  produced: 

Either  by  changing  the  position  angle  between  the  commuta- 
tor brushes  and  the  resultant  magnetic  flux,  so  that  the  direct 
voltage  between  the  brushes  is  not  the  maximum  diametrical 
alternating  voltage  but  only  a  part  thereof. 

Or  by  changing  the  maximum  diametrical  alternating  voltage, 
at  constant  effective  impressed  voltage,  by  wave-shape  distortion 
by  the  superposition  of  higher  harmonics. 

In  the  former  case,  only  a  reduction  of  the  direct  voltage  be- 
low the  normal  value  can  be  produced,  while  in  the  latter  case 
an  increase  as  well  as  a  reduction  can  be  produced,  an  increase 
if  the  higher  harmonics  are  in  phase,  and  a  reduction  if  the  higher 
harmonics  are  in  opposition  to  the  fundamental  wave  of  the  dia- 
metrical or  Y  voltage. 

A.  Variable  Ratio  by  a  Change  of  the  Position  Angle  between 
Commutator  Brushes  and  Resultant  Magnetic  Flux 

231.  Let,  in  the  commutating  machine  shown  diagrammatic- 
ally  in  Fig.  195,  the  potential  difference,  or  alternating  voltage 
between  one  point,  a,  of  the  armature  winding  and  the  neutral,  0 
(that  is,  the  Y  voltage,  or  half  the  diametrical  voltage)  be  repre- 
sented by  the  sine  wave,  Fig.  197.     This  potential  difference  is 
a  maximum,  e,  when  a  stands  at  the  magnetic  neutral,  at  A  or  B. 

422 


REGULATING  POLE  CONVERTERS 


423 


If,  therefore,  the  brushes  are  located  at  the  magnetic  neutral, 
A  and  B,  the  voltage  between  the  brushes  is  the  potential  differ- 
ence between  A  and  B,  or  twice  the  maximum  Y  voltage,  2  e, 
as  indicated  in  Fig.  197.  If  now  the  brushes  are  shifted  by  an 
angle,  T,  to  position  C  and  D,  Fig.  196,  the  direct  voltage  between 


81 


s — 


FIG.  195. — Diagram  of 
commutating  machine 
with  brushes  in  the  mag- 
netic neutral. 


FIG.  196. — E.m.f.  variation 
by  shifting  the  brushes. 


the  brushes  is  the  potential  difference  between  C  and  D,  or  2  e 
cos  r  with  a  sine  wave.  Thus,  by  shifting  the  brushes  from  the 
position  A,  B,  at  right  angles  with  the  magnetic  flux,  to  the  posi- 
tion E>  F,  in  line  with  the  magnetic  flux,  any  direct  voltage  be- 


FIG.  197.— Sine  wave  of  e.m.f. 


tween  2  e  and  0  can  be  produced,  with  the  same  wave  of  alter- 
nating volage,  a. 

As  seen,  this  variation  of  direct  voltage  between  its  maximum 
value  and  zero,  at  constant  impressed  alternating  voltage,  is  in- 


424 


ELECTRICAL  APPARATUS 


dependent  of  the  wave  shape,  and  thus  can  be  produced  whether 
the  alternating  voltage  is  a  sine  wave  or  any  other  wave. 

It  is  obvious  that,  instead  of  shifting  the  brushes  on  the  com- 
mutator, the  magnetic  field  poles  may  be  shifted,  in  the  opposite 
direction,  by  the  same  angle,  as  shown  in  Fig.  198,  A,  B,  C. 

Instead  of  mechanically  shifting  the  field  poles,  they  can  be 
shifted  electrically,  by  having  each  field  pole  consist  of  a  number 
of  sections,  and  successively  reversing  the  polarity  of  these  sec- 
tions, as  shown  in  Fig.  199,  A,  J5,  C,  D. 


FIG.  198. — E.m.f.  variation  by  mechanically  shifting  the  poles. 

Instead  of  having  a  large  number  of  field  pole  sections,  obvi- 
ously two  sections  are  sufficient,  and  the  same  gradual  change 
can  be  brought  about  by  not  merely  reversing  the  sections  but 
reducing  the  excitation  down  to  zero  and  bringing  it  up  again  in 
opposite  direction,  as  shown  in  Fig.  200,  A,  B,  C,  D,  E. 


A  B  C  D 

FIG.  199. — E.m.f.  variation  by  electrically  shifting  the  poles. 

In  this  case,  when  reducing  one  section  in  polarity,  the  other 
section  must  be  increased  by  approximately  the  same  amount, 
to  maintain  the  same  alternating  voltage. 

When  changing  the  direct  voltage  by  mechanically  shifting 
the  brushes,  as  soon  as  the  brushes  come  under  the  field  pole 
faces,  self-inductive  sparking  on  the  commutator  would  result 
if  the  iron  of  the  field  poles  were  not  kept  away  from  the  brush 


REGULATING  POLE  CONVERTERS 


425 


position  by  having  a  slot  in  the  field  poles,  as  indicated  in  dotted 
line  in  Fig.  196  and  Fig.  198,  B.  With  the  arrangement  in  Figs. 
196  and  198,  this  is  not  feasible  mechanically,  and  these  arrange- 


FIG.  200. — E.m.f.  variation  by  shifting-flux  distribution. 

ments  are,  therefore,  unsuitable.  It  is  feasible,  however,  as 
shown  in  Figs.  199  and  200,  that  is,  when  shifting  the  resultant 
magnetic  flux  electrically,  to  leave  a  commutating  space  between 


FIG.  201. — Variable  ratio  or  split-pole  converter. 

the  polar  projections  of  the  field  at  the  brushes,  as  shown  in  Fig. 
200,  and  thus  secure  as  good  commutation  as  in  any  other  com- 
mutating machine. 


426  ELECTRICAL  APPARATUS 

Such  a  variable-ratio  converter,  then,  comprises  an  armature 
A,  Fig.  201,  with  the  brushes,  B,  B',  in  fixed  position  and  field 
poles,  P,  P',  separated  by  interpolar  spaces,  C,  C',  of  such  width  as 
required  for  commutation.  Each  field  pole  consists  of  two  parts, 
P  and  PI,  usually  of  different  relative  size,  separated  by  a  narrow 
space,  DD',  and  provided  with  independent  windings.  By  vary- 
ing, then,  the  relative  excitation  of  the  two  polar  sections,  P  and 
PI,  an  effective  shift  of  the  resultant  field  flux  and  a  corresponding 
change  of  the  direct  voltage  is  produced. 

As  this  method  of  voltage  variation  does  not  depend  upon  the 
wave  shape,  by  the  design  of  the  field  pole  faces  and  the  pitch 
of  the  armature  winding  the  alternating  voltage  wave  can  be 
made  as  near  a  sine  wave  as  desired.  Usually  not  much  atten- 
tion is  paid  hereto,  as  experience  shows  that  the  usual  distributed 
winding  of  the  commutating  machine  gives  a  sufficiently  close 
approach  to  sine  shape. 

Armature  Reaction  and  Commutation 

232.  With  the  brushes  in  quadrature  position  to  the  resultant 
magnetic  flux,  and  at  normal  voltage  ratio,  the  direct-current 
generator  armature  reaction  of  the  converter  equals  the  syn- 
chronous-motor armature  reaction  of  the  power  component  of 
the  alternating  current,  and  at  unity  power-factor  the  converter 
thus  has  no  resultant  armature  reaction,  while  with  a  lagging 
or  leading  current  it  has  the  magnetizing  or  demagnetizing  re- 
action of  the  wattless  component  of  the  current. 

If  by  a  shift  of  the  resultant  flux  from  quadrature  position 
with  the  brushes,  by  angle,  r,  the  direct  voltage  is  reduced  by 
factor  cos  T,  the  direct  current  and  therewith  the  direct-current 

armature  reaction  are  increased,  by  factor,  -  — ,  as  by  the  law 

COS  T 

of  conservation  of  energy  the  direct-current  output  must  equal 
the  alternating-current  input  (neglecting  losses).  The  direct- 
current  armature  reaction,  £,  therefore  ceases  to  be  equal  to  the 
armature  reaction  of  the  alternating  energy  current,  30,  but  is 

greater  by  f actor,          : 


COS  T 

The  alternating-current  armature  reaction,  SFo,  at  no  phase  dis- 
placement, is  in  quadrature  position  with  the  magnetic  flux. 


REGULATING  POLE  CONVERTERS  427 

The  direct-current  armature  reaction,  $,  however,  appears  in  the 
position  of  the  brushes,  or  shifted  against  quadrature  position 
by  angle  r;  that  is,  the  direct-current  armature  reaction  is  not  in 
opposition  to  the  alternating-current  armature  reaction,  but 
differs  therefrom  by  angle  T,  and  so  can  be  resolved  into  two 
components,  a  component  in  opposition  to  the  alternating-cur- 
rent armature  reaction,  50,  that  is,  in  quadrature  position  with 
the  resultant  magnetic  flux: 

<5"  =  $  cos  T  =  SFo, 

that  is,  equal  and  opposite  to  the  alternating-current  armature 
reaction,  and  thus  neutralizing  the  same;  and  a  component  in 
quadrature  position  with  the  alternating-current  armature  reac- 
tion, $0,  or  in  phase  with  the  resultant  magnetic  flux,  that  is, 
magnetizing  or  demagnetizing: 

&  =  $  sin  T  =  ^o  tan  T; 

that  is,  in  the  variable-ratio  converter  the  alternating-current 
armature  reaction  at  unity  power-factor  is  neutralized  by  a 
component  of  the  direct-current  armature  reaction,  but  a  result- 
ant armature  reaction,  $',  remains,  in  the  direction  of  the  resultant 
magnetic  field,  that  is,  shifted  by  angle  (90  —  r)  against  the 
position  of  brushes.  This  armature  reaction  is  magnetizing  or 
demagnetizing,  depending  on  the  direction  of  the  shift  of  the 
field,  T. 

It  can  be  resolved  into  two  components,  one  at  right  angles 
with  the  brushes: 

tf'i  =  $'  cos  T  =  5Q  sin  r, 
and  one,  in  line  with  the  brushes: 

$'2  =  £'  sin  T  ='  £  sin2  r  =  ^o  sin  r  tan  r, 

as  shown  diagrammatically  in  Figs.  202  and  203. 

There  exists  thus  a  resultant  armature  reaction  in  the  direc- 
tion of  the  brushes,  and  thus  harmful  for  commutation,  just  as 
in  the  direct-current  generator,  except  that  this  armature  reac- 
tion in  the  direction  of  the  brushes  is  only  $'2  =  &  sin2  T,  that  is, 
sin2  T  of  the  value  of  that  of  a  direct-current  generator. 

The  value  of  $'%  can  also  be  derived  directly,  as  the  difference 
between  the  direct-current  armature  reaction,  5,  and  the  com- 


428 


ELECTRICAL  APPARATUS 


5= - 


FIG.  202. — Diagram  of  m.m.fs.  in  split-pole  converter. 


FIG.  203. — Diagram  of  m.m.fs.  in  split-pole  converter. 


REGULATING  POLE  CONVERTERS  429 

ponent  of  the  alternating-current  armature  reaction,  in  the  direc- 
tion of  the  brushes,  $0  cos  r,  that  is : 

5^2  =  $  —  ^o  cos  T  =  $  (1  —  cos2  r)  =  $  sin2  r  =  30  sin  r  tan  r. 

233.  The  shift  of  the  resultant  magnetic  flux,  by  angle  r,  gives 
a  component  of  the  m.m.f.  of  field  excitation,  $"/  =  5y  sin  r, 
(where  5y  =  m.m.f.  of  field  excitation),  in  the  direction  of  the 
commutator  brushes,  and  either  in  the  direction  of  armature 
reaction,  thus  interfering  with  commutation,  or  in  opposition  to 
the  armature  reaction,  thus  improving  commutation. 

If  the  magnetic  flux  is  shifted  in  the  direction  of  armature 
rotation,  that  is,  that  section  of  the  field  pole  weakened  toward 
which  the  armature  moves,  as  in  Fig.  202,  the  component  $"/ 
of  field  excitation  at  the  brushes  is  in  the  same  direction  as  the 
armature  reaction,  $'2,  thus  adds  itself  thereto  and  impairs  the 
commutation,  and  such  a  converter  is  hardly  operative.  In  this 
case  the  component  of  armature  reaction,  $',  in  the  direction  of 
the  field  flux  is  magnetizing. 

If  the  magnetic  flux  is  shifted  in  opposite  direction  to  the 
armature  reaction,  that  is,  that  section  of  the  field  pole  weakened 
which  the  armature  conductor  leaves,  as  in  Fig.  203,  the  com- 
ponent, $"  f,  of  field  excitation  at  the  brushes  is  in  opposite  direc- 
tion to  the  armature  reaction,  $'2,  therefore  reverses  it,  if  suffi- 
ciently large,  and  gives  a  commutating  or  reversing  flux,  3>r,  that 
is,  improves  commutation  so  that  this  arrangement  is  used  in 
such  converters.  In  this  case,  however,  the  component  of  arma- 
ture reaction,  $',  in  the  direction  of  the  field  flux  is  demagnet- 
izing, and  with  increasing  load  the  field  excitation  has  to  be  in- 
creased by  ff'  to  maintain  constant  flux.  Such  a  converter  thus 
requires  compounding,  as  by  a  series  field,  to  take  care  of  the 
demagnetizing  armature  reaction. 

If  the  alternating  current  is  not  in  phase  with  the  field,  but 
lags  or  leads,  the  armature  reaction  of  the  lagging  or  leading 
component  of  current  superimposes  upon  the  resultant  armature 
reaction,  5',  and  increases  it — with  lagging  current  in  Fig.  202, 
leading  current  in  Fig.  203 — or  decreases  it — with  lagging  cur- 
rent in  Fig.  203,  leading  current  in  Fig.  202 — and  with  lag  of  the 
alternating  current,  by  phase  angle,  6  =  r\  under  the  conditions 
of  Fig.  203,  the  total  resultant  armature  reaction  vanishes,  that  is, 
the1  lagging  component  of  synchronous-motor  armature  reaction 
compensates  for  the  component  of  the  direct-current  reaction, 


430 


ELECTRICAL  APPARATUS 


which  is  not  compensate^  by  the  armature  reaction  of  the  power 
component  of  the  alternating  current:  It  is  interesting  to  note 
that  in  this  case,  in  regard  to  heating,  output  based  thereon,  etc., 
the  converter  equals  that  of  one  of  normal  voltage  ratio. 

B.  Variable  Ratio  by  Change  of  Wave  Shape  of  the  Y  Voltage 

234.  If  in  the  converter  shown  diagrammatically  in  Fig.  204 
the  magnetic  flux  disposition  and  the  pitch  of  the  armature 
winding  are  such  that  the  potential  difference  between  the  point, 

a,  of  the  armature  and  the  neutral  0, 
or  the  Y  voltage,  is  a  sine  wave,  Fig. 
205  A,  then  the  voltage  ratio  is 
normal.  Assume,  however,  that 
the  voltage  curve,  a,  differs  from 
sine  shape  by  the  superposition  of 
some  higher  harmonics:  the  third 
harmonic  in  Figs.  205  B  and  C; 
the  fifth  harmonic  in  Figs.  205  D 
and  E.  If,  then,  these  higher 
harmonics  are  in  phase  with  the 
fundamental,  that  is,  their  maxima 
coincide,  as  in  Figs.  205  B  and  D, 
they  increase  the  maximum  of  the 
alternating  voltage,  and  thereby  the 
direct  voltage ;  and  if  these  harmonics 
are  in  opposition  to  the  funda- 
mental, as  in  Figs.  205  C  and  E,  they  decrease  the  maximum 
alternating  and  thereby  the  direct  voltage,  without  appreciably 
affecting  the  effective  value  of  the  alternating  voltage.  For  in- 
stance, a  higher  harmonic  of  30  per  cent,  of  the  fundamental 
increases  or  decreases  the  direct  voltage  by  30  per  cent.,  but 
varies-  the  effective  alternating  voltage  only  by  \/l  -f  0.32  = 
1.044,  or  4.4  per  cent. 

The  superposition  of  higher  harmonics  thus  offers  a  means  of 
increasing  as  well  as  decreasing  the  direct  voltage,  at  constant 
alternating  voltage,  and  without  shifting  the  angle  between  the 
brush  position  and  resultant  magnetic  flux. 

Since,  however,  the  terminal  voltage  of  the  converter  does  not 
only  depend  on  the  generated  e.m.f.  of  the  converter,  but  also 
on  that  of  the  generator,  and  is  a  resultant  of  the  two  e.m.fs.  in 
approximately  inverse  proportion  to  the  impedances  from  the 
converter  terminals  to  the  two  respective  generated  e.m.fs.,  for 


FIG.  204. — Variable  ratio  con- 
verter by  changing  wave  shape 
of  the  Y.  e.m.f. 


REGULATING  POLE  CONVERTERS 


431 


varying  the  converter  ratio  only  such  higher  harmonics  can  be 
used  which  may  exist  in  the  Y  voltage  without  appearing  in  the 
converter  terminal  voltage  or  supply  voltage. 

In  general,  in  an  n-phase  system  an  nth  harmonic  existing  in 
the  star  or  Y  voltage  does  not  appear  in  the  ring  or  delta  voltage, 


FIG.  205. — Superposition  of  harmonics  to  change  the  e.m.f.  ratio. 

as  the  ring  voltage  is  the  combination  of  two  star  voltages  dis- 

1  SO 
placed  in  phase  by degrees  for  the  fundamental,  and  thus  by 

iL 

180°,  or  in  opposition,  for  the  nth  harmonic. 

Thus,  in  a  three-phase  system,  the  third  harmonic  can  be  in- 
troduced into  the  Y  voltage  of  the  converter,  as  in  Figs.  205  B 
and  C,  without  affecting  or  appearing  in  the  delta  voltage,  so 
can  be  used  for  varying  the  direct-current  voltage,  while  the  fifth 
harmonic  can  not  be  used  in  this  way,  but  would  reappear  and 


432 


ELECTRICAL  APPARATUS 


cause  a  short-circuit  current  in  the  supply  voltage,  hence  should 
be  made  sufficiently  small  to  be  harmless. 

235.  The  third  harmonic  thus  can  be  used  for  varying  the 
direct  voltage  in  the  three-phase  converter  diagrammatically 
shown  in  Fig.  206  A,  and  also  in  the  six-phase  converter  with 


FIG.  206. — Transformer  connections  for  varying  the  e.m.f.  ratio  by  super- 
position of  the  third  harmonic. 

double-delta  connection,  as  shown  in  Fig.  206  B,  or  double- Y 
connection,  as  shown  in  Fig.  206  C,  since  this  consists  of  two  sepa- 
rate three-phase  triangles  of  voltage  supply,  and  neither  of  them 
contains  the  third  harmonic.  In  such  a  six-phase  converter 
with  double- Y  connection,  Fig.  206  C,  the  two  neutrals,  however, 


REGULATING  POLE  CONVERTERS 


433 


must  not  be  connected  together,  as  the  third  harmonic  voltage 
exists  between  the  neutrals.  In  the  six-phase  converter  with 
diametrical  connections,  the  third  harmonic  of  the  Y  voltage  ap- 
pears in  the  terminal  voltage,  as  the  diametrical  voltage  is  twice 
the  Y  voltage.  In  such  a  converter,  if  the  primaries  of  the  sup- 


FIG.  207. — Shell-type  transformers. 

ply  transformers  are  connected  in  delta,  as  in  Fig.  206  D,  the 
third  harmonic  is  short-circuited  in  the  primary  voltage  triangle, 
and  thus  produces  excessive  currents,  which  cause  heating  and 
interfere  with  the  voltage  regulation,  therefore,  this  arrangement 


FIG.  208, — Core-type  transformer. 

is  not  permissible.  If,  however,  the  primaries  are  connected  in 
Y,  as  in  Fig.  206  E,  and  either  three  separate  single-phase  trans- 
formers, or  a  three-phase  transformer  with  three  independent 
magnetic  circuits,  is  used,  as  in  Fig.  207,  the  triple-frequency 
voltages  in  the  primary  are  in  phase  with  each  other  between 


28 


434 


ELECTRICAL  APPARATUS 


the  line  and  the  neutral,  and  thus,  with  isolated  neutral,  can  not 
produce  any  current.  With  a  three-phase  transformer  as  shown 
in  Fig.  208,  that  is,  in  which  the  magnetic  circuit  of  the  third 
harmonic  is  open,  triple-frequency  currents  can  exist  in  the  sec- 
ondary and  this  arrangement  therefore  is  not  satisfactory. 

In  two-phase  converters,  higher  harmonics  can  be  used  for 
regulation  only  if  the  transformers  are  connected  in  such  a  man- 
ner that  the  regulating  harmonic,  which  appears  in  the  converter 
terminal  voltage,  does  not  appear  in  the  transformer  terminals, 
that  is,  by  the  connection  analogous  to  Figs.  206  E  and  207. 

Since  the  direct-voltage  regulation  of  a  three-phase  or  six- 
phase  converter  of  this  type  is  produced  by  the  third  harmonic, 


FIG.  209. — Y  e.m.f.  wave. 


the  problem  is  to  design  the  magnetic  circuit  of  the  converter 
so  as  to  produce  the  maximum  third  harmonic,  the  minimum 
fifth  and  seventh  harmonics. 

If  q  =  interpolar  space,  thus  (1  —  q)  =  pole  arc,  as  fraction 
of  pitch,  the  wave  shape  of  the  voltage  generated  between  the 
point,  a,  of  a  full-pitch  distributed  winding — as  generally  used 
for  commutating  machines — and  the  neutral,  or  the  induced  Y 
voltage  of  the  system  is  a  triangle  with  the  top  cut  off  for  dis- 
tance q,  as  shown  in  Fig.  209,  when  neglecting  magnetic  spread 
at  the  pole  corners. 

If  then  eo  =  voltage  generated  per  armature  turn  while  in 
front  of  the  field  pole  (which  is  proportional  to  the  magnetic  den- 
sity in  the  air  gap),  m  =  series  turns  from  brush  to  brush,  the 
maximum  voltage  of  the  wave  shown  in  Fig.  209  is: 

E0  =  me0  (1  —  q)} 

developed  into  a  Fourier  series,  this  gives,  as  the  equation  of  the 
voltage  wave  a,  Fig.  188: 


~(2w  -  1): 


cos  (2  n  -  1)6', 


REGULATING  POLE  CONVERTERS  435 

or,  substituting  for  EQ,  and  denoting: 

7T2 

2n  -  1 
"     COS 2 —  qir 

e=A^n    (2n-l)'      ""E"-!)' 
=  A  j  cos  g  -  cos  0  +  Q  cos  3  5  ~  cos  30  +  ^  cos  5  <?  -  cos  5  0 

I  £  \y  £  ^0  ^ 

-j 

-f  VQ  cos  7  q  |  cos  7  0  -\ 

Thus  the  third  harmonic  is  a  positive  maximum  for  q  =  0,  or 
100  per  cent,  pole  arc,  and  a  negative  maximum  for  q  =  %,  or 
33.3  per  cent,  pole  arc. 

For  maximum  direct  voltage,  q  should  therefore  be  made  as 
small,  that  is,  the  pole  arc  as  large,  as  commutation  permits. 
In  general,  the  minimum  permissible  value  of  q  is  about  0.15  to 
0.20. 

The  fifth  harmonic  vanishes  for  q  =  0.20  and  q  =  0.60,  and 
the  seventh  harmonic  for  q  =  0.143,  0.429,  and  0.714. 

For  small  values  of  q,  the  sum  of  the  fifth  and  seventh  har- 
monics is  a  minimum  for  about  q  =  0.18,  or  82  per  cent,  pole  arc. 
Then  for  q  =  0.18,  or  82  per  cent,  pole  arc: 

ei  =  A  {0.960  cos  0  +  0.0736  cos  3  0  +  0.0062  cos  5  0 

-  0.0081  cos  7  0  +    .    .    .  j 

=  0.960  A  {cos  0  +  0.0766  cos  3  0  +  0.0065  cos  5  0 

-  0.0084  cos  7  0  +  .  .  .  j ; 

that  is,  the  third  harmonic  is  less  than  8  per  cent.,  so  that  not 
much  voltage  rise  can  be  produced  in  this  manner,  while  the 
fifth  and  seventh  harmonics  together  are  only  1.3  per  cent.,  thus 
negligible. 

236.  Better  results  are  given  by  reversing  or  at  least  lowering 
the  flux  in  the  center  of  the  field  pole.  Thus,  dividing  the  pole 
face  into  three  equal  sections,  the  middle  section,  of  27  per  cent, 
pole  arc,  gives  the  voltage  curve,  q  =  0.73,  thus: 

e2  =  A  {0.411  cos  0  -  0.1062  cos  3  0  +  0.0342  cos  5  0 

-0.0035  cos  70   .    .    .} 

=  0.411  A  {cos  0  -  0.258    cos  3  0  +  0.083  cos  5  0 
-0.0085  cos  7  0   .    .    .}• 

The  voltage  curves  given  by  reducing  the  pole  center  to  one- 


436  ELECTRICAL  APPARATUS 

half  intensity,  to  zero,  reversing  it  to  half  intensity,  to  full  in- 
tensity, and  to  such  intensity  that  the  fundamental  disappears, 
then  are  given  by: 

Center  part 
of  pole 
density 

(1)  full,   e  =  ei  =0.960  A  {cos  0+0.077  cos  3  0 

+0.0065  cos  5  0-0.0084  cos  76.    .    . } 

(2)  0.5,    e  =  ei-0.5  e2   =0.755  A  {cos  (9+0.168  cos  3  6 

-0.0144  cos  5  0-0.0085  cos  76.    .    .} 

(3)  0,       e  =  ei-e2          =0.549  A  {cos  5+0.328  cos  3  6 

-  0.053  cos  5  B  -  0.084  cos  7  6 .    .    .  1 

(4)  -0.5     e  =  ei-1.5e2   =0.344  A  {cos  6  +0.680  cos  3  6 

-0.131  cos  5  0-0.0084  cos  7e.    .    .} 

(5)  -  full,  e  =  el -2  e2      =0.138  A  {cos  (9+2.07  cos  3  B 

-0.4500856*  -0.008  cos  7  e.    .    .} 

(6)  -1.17,  e  =  ei-2.34  ^2  =  0.322  A  {cos  3  0-0.227  cos  50.    .    .). 

It  is  interesting  to  note  that  in  the  last  case  the  fundamental 
frequency  disappears  and  the  machine  is  a  generator  of  triple 
frequency,  that  is,  produces  or  consumes  a  frequency  equal  to 
three  times  synchronous  frequency.  In  this  case  the  seventh 
harmonic  also  disappears,  and  only  the  fifth  is  appreciable,  but 
could  be  greatly  reduced  by  a  different  kind  of  pole  arc.  From 
above  table  follows : 

(1)        (2)        (3)'       (4)        (5)    (6)       normal 
Maximum  funda- 


mental    alter- 


0.960   0.755   0.549   0.344   0.138   0  0.960 


nating  volts . . . 
Direct  volts 1.033    0.883    0.743    0.578    0.423    0.322    0.960 

237.  It  is  seen  that  a  considerable  increase  of  direct  voltage 
beyond  the  normal  ratio  involves  a  sacrifice  of  output,  due  to 
the  decrease  or  reversal  of  a  part  of  the  magnetic  flux,  whereby 
the  air-gap  section  is  not  fully  utilized.  Thus  it  is  not  advisable 
to  go  too  far  in  this  direction. 

By  the  superposition  of  the  third  harmonic  upon  the  funda- 
mental wave  of  the  Y  voltage,  in  a  converter  with  three  sections 
per  pole,  thus  an  increase  of  direct  voltage  over  its  normal 
voltage  can  be  produced  by  lowering  the  excitation  of  the  middle 
section  and  raising  that  of  the  outside  sections  of  the  field  pole, 
and  also  inversely  a  decrease  of  the  direct  voltage  below  its 
normal  value  by  raising  the  excitation  of  the  middle  section 


REGULATING  POLE  CONVERTERS  437 

and  decreasing  that  of  the  outside  sections  of  the  field  poles; 
that  is,  in  the  latter  case  making  the  magnetic  flux  distribution 
at  the  armature  periphery  peaked,  in  the  former  case  by  making 
the  flux  distribution  flat-topped  or  even  double-peaked. 

Armature  Reaction  and  Commutation 

238.  In  such  a  split-pole  converter  let  p  equal  ratio  of  direct 
voltage  to  that  voltage  which  it  would  have,  with  the  same 
alternating  impressed  voltage,  at  normal  voltage  ratio,  where 
p  >  1  represents  an  overnormal,  p  <  1  a  subnormal  direct 
voltage.  The  direct  current,  and  thereby  the  direct-current 
armature  reaction,  then  is  changed  from  the  value  which  it 

would  have  at  normal  voltage  ratio,  by  the  factor  —  ,  as  the 

product  of  direct  volts  and  amperes  must  be  the  same  as  at 
normal  voltage  ratio,  being  equal  to  the  alternating  power 
input  minus  losses. 

With  unity  power-factor,  the  direct-current  armature  reac- 
tion, 3,  in  a  converter  of  normal  voltage  ratio  is  equal  and  opposite, 
and  thus  neutralized  by  the  alternating-current  armature  reac- 
tion, $0,  and  at  a  change  of  voltage  ratio  from  normal,  by  factor 

p,  and  thus  change  of  direct  current  by  factor  —  The  direct- 
current  armature  react  'on  thus  is: 


P 
hence,  leaves  an  uncompensated  resultant. 

As  the  alternating-current  armature  reaction  at  unity  power- 
factor  is  in  quadrature  with  the  magnetic  flux,  and  the  direct- 
current  armature  reaction  in  line  with  the  brushes,  and  with 
this  type  of  converter  the  brushes  stand  at  the  magnetic  neutral, 
that  is,  at  right  angles  to  the  magnetic  flux,  the  two  armature 
reactions  are  in  the  same  direction  in  opposition  with  each  other, 
and  thus  leave  the  resultant,  in  the  direction  of  the  commutator 
brushes  : 


The  converter  thus  has  an  armature  reaction  proportional  to 
the  deviation  of  the  voltage  ratio  from  normal. 

239.  If  p  >  1,  or  overnormal   direct  voltage,  the   armature 


438 


ELECTRICAL  APPARATUS 


reaction  is  negative,  or  motor  reaction,  and  the  magnetic  flux 
produced  by  it  at  the  commutator  brushes  thus  a  commutating 
flux.  If  p  <  1,  or  subnormal  direct  voltage,  the  armature 
reaction  is  positive,  that  is,  the  same  as  in  a  direct-current  gen- 
erator, but  less  in  intensity,  and  thus  the  magnetic  flux  of  arma- 
ture reaction  tends  to  impair  commutation.  In  a  direct-current 
generator,  by  shifting  the  brushes  to  the  edge  of  the  field  poles, 
the  field  flux  is  used  as  reversing  flux  to  give  commutation.  In 
this  converter,  however,  decrease  of  direct  voltage  is  produced  by 
lowering  the  outside  sections  of  the  field  poles,  and  the  edge  of 
the  field  may  not  have  a  sufficient  flux  density  to  give  commuta- 


t- 


FIG.  210.  —  Three-section  pole  for  variable-ratio  converter. 

tion,  with  a  considerable  decrease  of  voltage  below  normal,  and 
thus  a  separate  commutating  pole  is  required.  Preferably  this 
type  of  converter  should  be  used  only  for  raising  the  voltage, 
for  lowering  the  voltage  the  other  type,  which  operates  by  a 
shift  of  the  resultant  flux,  and  so  gives  a  component  of  the  main 
field  flux  as  commutating  flux,  should  be  used,  or  a  combination 
of  both  types. 

With  a  polar  construction  consisting  of  three  sections,  this 
can  be  done  by  having  the  middle  section  at  low,  the  outside 
sections  at  high  excitation  for  maximum  voltage,  and,  to  de- 
crease the  voltage,  raise  the  excitation  of  the  center  section,  but 
instead  of  lowering  both  outside  sections,  leave  the  section  in  the 
direction  of  the  armature  rotation  unchanged,  while  lowering 
the  other  outside  section  twice  as  much,  and  thus  produce,  in 
addition  to  the  change  of  wave  shape,  a  shift  of  the  flux,  as 
represented  by  the  scheme  Fig.  210. 


Pole  section   . 
Max.  voltage 


.     1 
.  +(B 


Magnetic  Density 

2  3  .  1' 

0         +(B  -(B 


2' 

O 


0 


+(B        +(B 


Min.  voltage  .  .  .  - 


3' 

-(B 

-(B 


0        -(B         -(B 

-(B 


REGULATING  POLE  CONVERTERS 


439 


Where  the  required  voltage  range  above  normal  is  not  greater 
than  can  be  produced  by  the  third  harmonic  of  a  large  pole  arc 
with  uniform  density,  this  combination  of  voltage  regulation 
by  both  methods  can  be  carried  out  with  two  sections  of  the 
field  poles,  of  which  the  one  (toward  which  the  armature  moves) 
is  greater  than  the  other,  as  shown  in  Fig.  211,  and  the  variation 
then  is  as  follows: 


Pole  section   , 
Max.  voltage 


Min.  voltage 


1 
+       & 


Magnetic  Density 
2  1' 

(B  -     (B 


0 


V 

-  (B 
-1KCB 
-1^(B 
-1%(B 


FIG.  211. — Two-section  pole  for  variable-ratio  converter. 

Heating  and  Rating 

240.  The  distribution  of  current  in  the  armature  conductors 
of  the  variable-ratio  converter,  the  wave  form  of  the  actual 
or  differential  current  in  the  conductors,  and  the  effect  of  the 
wattless  current  thereon,  are  determined  in  the  same  manner  as 
in  the  standard  converter,  and  from  them  are  calculated  the  local 
heating  in  the  individual  armature  turns  and  the  mean  armature 
heating. 

In  an  n-phase  converter  of  normal  voltage  ratio,  let  EQ  = 
direct  voltage;  70  =  direct  current;  E°  =  alternating  voltage 
between  adjacent  collector  rings  (ring  voltage),  and  7°  =  alter- 
nating current  between  adjacent  collector  rings  (ring  current); 
then,  as  seen  in  the  preceding: 


E°  = 


E0  sin  - 
n 


(1) 


V  * 

and  as  by  the  law  of  conservation  of  energy,  the  output  must 
equal  the  input,  when  neglecting  losses : 


7°  = 


/o 


n  sin- 
n 


(2) 


440  ELECTRICAL  APPARATUS 

where  7°  is  the  power  component  of  the  current  corresponding 
to  the  direct-current  output. 

The  voltage  ratio  of  a  converter  can  be  varied: 
(a)  By    the    superposition    of    a    third    harmonic    upon    the 
tar  voltage,  or  diametrical  voltage,  which  does  not  appear  in 
he  ring  voltage,  or  voltage  between  the  collector  rings  of  the 
converter. 

(6)  By  shifting  the  direction  of  the  magnetic  flux. 

(a)  can  be  used  for  raising  the  direct  voltage  as  well  as  for 
lowering  it,  but  is  used  almost  always  for  the  former  purpose, 
since  when  using  this  method  for  lowering  the  direct  voltage 
commutation  is  impaired. 

(b)  can  be  used  only  for  lowering  the  direct  voltage. 

It  is  possible,  by  proportioning  the  relative  amounts  by  which 
the  two  methods  contribute  to  the  regulation'  of  the  voltage, 
to  maintain  a  proper  commutating  field  at  the  brushes  for  all 
loads  and  voltages.  Where,  however,  this  is  not  done,  the 
brushes  are  shifted  to  the  edge  of  the  next  field  pole,  and  into 
the  fringe  of  its  field,  thus  deriving  the  commutating  field. 

241.  In  such  a  variable-ratio  converter  let,  then,  t  =  intensity 
of  the  third  harmonic,  or  rather  of  that  component  of  it  which 
is  in  line  with  the  direct-current  brushes,  and  thus  does  the 
voltage  regulation,  as  fraction  of  the  fundamental  wave,  t  is 
chosen  as  positive  if  the  third  harmonic  increases  the  maximum 
of  the  fundamental  wave  (wide  pole  arc)  and  thus  raises  the 
direct  voltage,  and  negative  when  lowering  the  maximum  of  the 
fundamental  and  therewith  the  direct  voltage  (narrow  pole  arc). 

pi  =  loss  of  power  in  the  converter,  which  is  supplied  by  the 
current  (friction  and  core  loss)  as  fraction  of  the  alternating 
input  (assumed  as  4  per  cent,  in  the  numerical  example). 

n  =  angle  of  brush  shift  on  the  commutator,  counted  positive 
in  the  direction  of  rotation. 

6 'i  =  angle  of  time  lag  of  the  alternating  current  (thus  negative 
for  lead). 

ra  =  angle  of  shift  of  the  resultant  field  from  the  position  at 
right  angles  to  the  mechanical  neutral  (or  middle  between  the 
pole  corners  of  main  poles  and  auxiliary  poles),  counted  positive 
in  the  direction  opposite  to  the  direction  of  armature  rotation, 
that  is,  positive  in  that  direction  in  which  the  field  flux  has  been 
shifted  to  get  good  commutation,  as  discussed  in  the  preceding 
article. 


REGULATING  POLE  CONVERTERS  441 

Due  to  the  third  harmonic,  t,  and  the  angle  of  shift  of  the  field 
flux,  r0,  the  voltage  ratio  differs  from  the  normal  by  the  factor: 

(1    -M)   COSTa, 

and  the  ring  voltage  of  the  converter  thus  is: 


(1  +  t)  COS  r.' 
hence,  by  (1): 


. 

\/2  (1   +  0  COSTa 

and  the  power  component  of  the  ring  current  corresponding  to 
the  direct-current  output  thus  is,  when  neglecting  losses,  from 
(2): 

/'  =  1°  (1  +  0  cos  TO 

_  /o  A/2  (1+0  cos  Tgm  /\ 


TT 

n  sin  - 
n 


Due  to  the  loss,  pi,  in  the  converter,  this  current  is  increased  by 
(1  +  PJ)  in  a  direct  converter,  or  decreased  by  the  factor 
(1  —pi)  in  an  inverted  converter. 

The  power  component  of  the  alternating  current  thus  is: 

/!  =  r  (i  +  Pl) 


n  sin  - 


where  pi  may  be  considered  as  negative  in  an  inverted  converter. 
With  the  angle  of  lag  61,  the  reactive  component  of  the  current 
is: 

72  =  1  1  tan  0i, 


and  the  £ota£  alternating  ring  current  is  : 


Jo  V^  (1+Q  (l+P/)cosr 


n  sin  -  cos  0i 

n 


442 


ELECTRICAL  APPARATUS 


or,  introducing  for  simplicity  the  abbreviation: 

,  (1  +  t)  (1  +  pi)  COS  Ta 


it  is: 


cos  61 


I  = 


I0k\/2 

n  sin  - 
n 


(8) 
(9) 


242.  Let,  in  Fig.  212,  A'OA  represent  the  center  line  of  the 
magnetic  field  structure. 

The  resultant  magnetic  field  flux,  0$,  then  leads  OA  by  angle 
3>OA  =  ra. 

The  resultant  m.m.f .  of  the  alternating  power  current,  7i,  is  07i, 


FIG.  212. — Diagram  of  variable  ratio  converter. 

at  right  angles  to  0$,  and  the  resultant  m.m.f.  of  the  alternating 
reactive  current,  72,  is  Oh,  in  opposition  to  Oi>,  while  the  total 
alternating  current,  /,  is  01,  lagging  by  angle  0i  behind  O/i. 

The  m.m.f.  of  direct-current  armature  reaction  is  in  the  direc- 
tion of  the  brushes,  thus  lagging  by  angle  rh  behind  the  position 
OB,  where  BOA  =  90°,  and  given  by  0/0. 

The  angle  by  which  the  direct-current  m.m.f.,  O/o,  lags  in  space 
behind  the  total  alternating  m.m.f.,  07,  thus  is,  by  Fig.  212: 

T0.=    01    -  Ta   -  Tb.  (10) 

If  the  alternating  m.m.f.  in  a  converter  coincides  with  the 
direct-current  m.m.f.,  the  alternating  current  and  the  direct  cur- 
rent are  in  phase  with  each  other  in  the  armature  coil  midway 


REGULATING  POLE  CONVERTERS 


443 


between  adjacent  collector  rings,  and  the  current  heating  thus 
a  minimum  in  this  coil. 

Due  to  the  lag  in  space,  by  angle  TO,  of  the  direct-current 
m.m.f.  behind  the  alternating  current  m.m.f.,  the  reversal  of  the 
direct  current  is  reached  in  time  before  the  reversal  of  the  alter- 
nating current  in  the  armature  coil;  that  is,  the  alternating 
current  lags  behind  the  direct  current  by  angle,  00  =  TO,  in  the 


FIG.  213. — Alternating  and  direct  current  in  a  coil  midway  between 
adjacent  collector  leads. 

armature  coil  midway  between  adjacent  collector  leads,  as 
shown  by  Fig.  213,  and  in  an  armature  coil  displaced  by  angle,  T, 
from  the  middle  position  between  adjacent  collector  leads  the 
alternating  current  thus  lags  behind  the  direct  current  by  angle 
(r  +  00),  where  r  is  counted  positive  in  the  direction  of  armature 
rotation  (Fig.  214). 


FIG.  214. — Alternating  and  direct  current  in  a  coil  at  the  angle  T  from  the 

middle  position. 

The  alternating  current  in  armature  coil,  T,  thus  can  be  ex- 
pressed by: 

i  =  /V2sin  (e  -  T  -  00);  (11) 

hence,  substituting  (9): 


2/0/b    sin(0-T-00), 


n  sin- 
n 


(12) 


and  as  the  direct  current  in  this  armature  coil  is  -~t  and  opposite 


444 


ELECTRICAL  APPARATUS 


to  the  alternating  current,  i,  the  resultant  current  in  the  arma- 
ture coil,  T,  is: 


/o 

2 


.       TT 

n  sin  - 
n 


sin  (0  —  T  —  00)  —  1 


(13) 


and  the  ratio  of  heating,  of  the  resultant  current,  i0,  compared 
with  the  current,  ^,  of  the  same  machine  as  direct-current  gen- 
erator of  the  same  output,  thus  is : 

V«2  f         A-  If  1  2 

(e  -  r  -  00)  -  : 


n  sin  - 
n 


(14) 


Averaging  (14)  over  one  half  wave  gives  the  relative  heating 
of  the  armature  coil,  T,  as: 


1 

TT    =    ~ 


«tt 


1 


4.  fr 


fr  1   2 

-  sin  (0  -  r  -  tf0)  -  1     d».  (15) 


Integrated,  this  gives: 


^ 

TT    —  I      1 


16fccos( 


sin 


TT  n  sin  - 


(16) 


243.  Herefrom  follows  the  local  heating  in   any    armature 

7T 

coil,  T,  in  the  coils  adjacent  to  the  leads  by  substituting  T  =  +  - , 

i\> 

and  also  follows  the  average  armature  heating  by  averaging 


7T  from  r  =  —  -  to  r  =  H —  • 
n  n 

The  average  armature  heating  of  the  n-phase  converter  there- 
fore is: 


r  -  */+> 


or,  integrated: 


r  =  -- 


_ 

1       * 


cos  00 


_     .     0 

2       2 


n  sm 


(17) 


n 


REGULATING  POLE  CONVERTERS  445 

This  is  the  same  expression  as  found  for  the  average  armature 
heating  of  a  converter  of  normal  voltage  ratio,  when  operating 
with  an  angle  of  lag,  60,  of  the  alternating  current,  where  k  denotes 
the  ratio  of  the  total  alternating  current  to  the  alternating 
power  current  corresponding  to  the  direct-current  output. 

In  an  n-phase  variable  ratio  converter  (split-pole  converter), 
the  average  armature  heating  thus  is  given  by: 

8fc2  16  k  cos  0o 

7^  ~^2~ 

nL  suV  - 
n 

where 

,      _    (1   +  Q    (1  +  Pl)  COS  Ta         (    . 

~ 


l    —  Ta   — 


(10) 


and  t  =  ratio  of  third  harmonic  to  fundamental  alternating 
voltage  wave;  pi  —  ratio  of  loss  to  output;  0i  =  angle  of  lag  of 
alternating  current;  ra  =  angle  of  shift  of  the  resultant  mag- 
netic field  in  opposition  to  the  armature  rotation,  and  rb  =  angle 
of  shift  of  the  brushes  in  the  direction  of  the  armature  rotation. 
244.  For  a  three-phase  converter,  equation  (18)  gives  (n  =  3): 


r  =  ^~-  +  1  -  1-621  k  cos  0o  nm 

£1  (i\J) 

=  1.185  k2  +  1  -  1.621  k  cos  00. 
For  a  six-phase  converter,  equation  (18)  gives  (n  =  6): 


9 


+  1  -  1.621  k  cos 


=  0.889  k2  +  1  -  1.621  k  cos 
For  a  converter  of  normal  voltage  ratio: 

t  =  0,  Ta  =  0, 

using  no  brush  shift: 

rb  =  0; 

when  neglecting  the  losses: 

Pi  =  0, 
it  is : 

1 


(20) 


k  = 


COS  0i 

}      .  -    Q 
'0    —    "ij 


446  ELECTRICAL  APPARATUS 

and  equations  (19)  and  (20)  assume  the  form: 
Three-phase : 


-  0.621. 
Six-phase : 

T  =  (?^y  -  0.621. 

The  equation  (18)  is  the  most  general  equation  of  the  relative 
heating  of  the  synchronous  converter,  including  phase  displace- 
ment, 0i,  losses,  pi,  shift  of  brushes,  rb,  shift  of  the  resultant  mag- 
netic flux,  ra,  and  the  third  harmonic,  t. 

While  in  a  converter  of  standard  or  normal  ratio  the  armature 
heating  is  a  minimum  for  unity  power-factor,  this  is  not  in  gen- 
eral the  case,  but  the  heating  may  be  considerably  less  at  same 
lagging  current,  more  at  leading  current,  than  at  unity  power- 
factor,  and  inversely. 

245.  It  is  interesting  therefore  to  determine  under  which  con- 
ditions of  phase  displacement  the  armature  heating  is  a  minimum 
so  as  to  use  these  conditions  as  far  as  possible  and  avoid  con- 
ditions differing  very  greatly  therefrom,  as  in  the  latter  case 
the  armature  heating  may  become  excessive. 

Substituting  for  k  and  00  from  equations  (8)  and  (10)  into 
equation  (18)  gives: 


=  1   | 


n2  sin2  -  cos2  0j 

n 

16  (1    +   t)    (1    +    Pl)    COS  Ta  COS    (0f    —   Ta    —   Tb)          /inv 

,*  cos  9l  '    (19) 


Substituting : 


-  sin  -  =  m,        (20) 

7T  Tl 


which  is  a  constant  of  the  converter  type,  and  is  for  a  three- 
phase  converter,  w3  =  0.744;  for  a  six-phase  converter,  w6  = 
0.955;  and  rearranging,  gives: 

r  _  1+  8    U+0'(l +  *)•«»•,. 


ra2 

"I  & 

2    (1    +  0   (1    +  Pi)    COS  Ta  COS  (ra  + 


REGULATING  POLE  CONVERTERS  447 

8    (1  +  Q«  (!  +  ,.)•  cos*  r. 

7T2  ra2 

1  R. 

-  —2  (1  +  0  (1  +  pi)  cos  ra  sin  (Ta  +  n)  tan  0^  (21) 

T  is  a  minimum  for  the  value,  61,  of  the  phase  displacement 
given  by: 


d  tan  0! 
and  this  gives,  differentiated  : 

ton  a  m*  sin  (Ta  +  Tfc) 

tan  c/2  =  7^—  —  ,\  /-i       •~~\  —     —  • 

(1    +  0   (1   +pi)  COST0 

Equation  (22)  gives  the  phase  angle,  62,  for  which,  at  given 
Ta,  n,  t  and  pij  the  armature  heating  becomes  a  minimum. 

Neglecting  the  losses,  pi,  if  the  brushes  are  not  shifted,  Tb  =  0, 
and  no  third  harmonic  exists,  t  =  0: 

tan  0'2  =  m2  tan  ra, 

where  m2  =  0.544  for  a  three-phase,  0.912  for  a  six-phase 
converter. 

For  a  six-phase  converter  it  thus  is  approximately  0'2  =.  ra, 
that  is,  the  heating  of  the  armature  is  a  minimum  if  the  alter- 
nating current  lags  by  the  same  angle  (or  nearly  the  same  angle) 
as  the  magnetic  flux  is  shifted  for  voltage  regulation. 

From  equation  (22)  it  follows  that  energy  losses  in  the  con- 
verter reduce  the  lag,  02,  required  for  minimum  heating;  brush 
shift  increases  the  required  lag;  a  third  harmonic,  t,  decreases 
the  required  lag  if  additional,  and  increases  it  if  subtractive. 

Substituting  (22)  into  (21)  gives  the  minimum  armature  heat- 
ing of  the  converter,  which  can  be  produced  by  choosing  the 
proper  phase  angle,  02,  for  the  alternating  current.  It  is  then, 
after  some  transpositions: 


r.v 

J 


TTM  L  m 

cos  Ta  cos  (TO  +  Tb)  —  m2  sin2  (ra  +  T&)  f 


The  term  "To  contains   the  constants  t,  pi,  Ta,  n  only  in  the 
square  under  the  bracket  and  thus  becomes  a  minimum  if  this 


448  ELECTRICAL  APPARATUS 

square  vanishes,  that  is,  if  between  the  quantities  t,  ph  ra,  n  such 
relations  exist  that: 

(1  +Q(1+  pi)  COS  ra 

—  Y~  --  cos  (Ta  +  Tb)  =  0.  (24) 

fit 

246.  Of  the  quantities  t,  pi,  ra,  rb;  pi  and  T&  are  determined 
by  the  machine  design,  t  and  ra,  however,  are  equivalent  to 
each  other,  that  is,  the  voltage  regulation  can  be  accomplished 
either  by  the  flux  shift,  ra,  or  by  the  third  harmonic,  t,  or  by  both, 
and  in  the  latter  case  can  be  divided  between  ra  and  t  so  as  to 
give  any  desired  relations  between  them. 

Equation  (24)  gives: 

_  ?ft2    COS      (jq    +     Tb) 


— 

(1    +   Pl   COS  Ta) 

and  by  choosing  the  third  harmonic,  t,  as  function  of  the  angle  of 
flux  shift  TO,  by  equation  (25),  the  converter  heating  becomes  a 
minimum,  and  is  : 

|1   '     •       i||  r,«  =  i-«£;  i  (26) 

hence: 

r0°  =  0.551  for  a  three-phase  converter,  (2.7) 

r0°  =  0.261  for  a  six-phase  converter.  (28) 

Substituting  (25)  into  (22)  gives: 

tan  62  =  tan  (ra  -f  rb)  ; 
hence  : 

82  =  ra  +  r6;  (29) 

or,  in  other  words,  the  converter  gives  minimum  heating  r0°  if 
the  angle  of  lag,  02,  equals  the  sum  of  the  angle  of  flux  shift,  ra,  and 
of  brush  shift,  T&. 

It  follows  herefrom  that,  regardless  of  the  losses,  pi,  of  the 
brush  shift,  n,  and  of  the  amount  of  voltage  regulation  required, 
that  is,  at  normal  voltage  ratio  as  well  as  any  other  ratio,  the 
same  minimum  converter  heating  r0°  can  be  secured  by  dividing 
the  voltage  regulation  between  the  angle  of  flux  shift,  ra,  and 
the  third  harmonic,  t,  in  the  manner  as  given  by  equation  (25), 
and  operating  at  a  phase  angle  between  alternating  current  and 
voltage  equal  to  the  sum  of  the  angles  of  flux  shift,  ra,  and  of  brush 
shift  n',  that  is,  the  heating  of  the  split-pole  converter  can  be 
made  the  same  as  that  of  the  standard  converter  of  normal 
voltage  ratio. 


REGULATING  POLE  CONVERTERS 


449 


.Choosing  pt  =  0.04,  or  4  per  cent,  loss  of  current,  equation 
(25)  gives,  for  the  three-phase  and  for  the  six-phase  converter: 
(a)  no  brush  shift  (T&  =  0) : 

$,o  =  o.467,  (30) 

J«°  =  0.123; 

that  is,  in  the  three-phase  converter  this  would  require  a  third 
harmonic  of  46.7  per  cent.,  which  is  hardly  feasible;  in  the  six- 
phase  converter  it  requires  a  third  harmonic  of  12.3  per  cent., 
which  is  quite  feasible. 

(6)  20°  brush  shift  (rb  =  20) :  . 

cos  (TO  +  T6) 


=  1  -  0.533 


COS  Ta 


for 


Since 


;6o  =  i  _  o.877  - 

COS  Ta 

=  0,  or  no  flux  shift,  this  gives : 
£300  =  0.500, 
£600  =  0.176. 

COS  (TO,  + 


n) 


(31) 


(32) 


<  1    for    brush    shift   in  the  direction  of 


COSra 

armature  rotation,  it  follows  that  shifting  the  brushes  increases 
the  third  harmonic  required  to  carry  out  the  voltage  regulation 
without  increase  of  converter  heating,  and  thus  is  undesirable. 

It  is  seen  that  the  third  harmonic,  t,  does  not  change  much 
with  the  flux  shift,  ra,  but  remains  approximately  constant,  and 
positive,  that  is,  voltage  raising. 

It  follows  herefrom  that  the  most  economical  arrangement 
regarding  converter  heating  is  to  use  in  the  six-phase  converter 
a  third  harmonic  of  about  17  to  18  per  cent,  for  raising  the  vol- 
tage (that  is,  a  very  large  pole  arc),  and  then'  do  the  regulation 
by  shifting  the  flux,  by  the  angle,  r0,  without  greatly  reducing  the 
third  harmonic,  that  is,  keep  a  wide  pole  arc  excited. 

As  in  a  three-phase  converter  the  required  third  harmonic  is 
impracticably  high,  it  follows  that  for  variable  voltage  ratio  the 
six-phase  converter  is  preferable,  because  its  armature  heating 
•  can  be  maintained  nearer  the  theoretical  minimum  by  propor- 
tioning t  and  ra. 


2'J 


CHAPTER  XXII 

UNIPOLAR  MACHINES 

Homopolar  or  Acyclic  Machines 

247.  If  a  conductor,  C,  revolves  around  one  pole  of  a  stationary 
magnet  shown  as  NS  in  Fig.  215,  a  continuous  voltage  is  induced 
in  the  conductor  by  its  cutting  of  the  lines  of  magnetic  force  of 
the  pole,  N,  and  this  voltage  can  be  supplied  to  an  external  cir- 
cuit, D,  by  stationary  brushes,  B\  and  Bz,  bearing  on  the  ends 
of  the  revolving  conductor,  C. 

The  voltage  is: 

e  =  /$  10-8, 

where  /  is  the  number  of  revolutions  per  second,  <£  the  magnetic 
flux  of  the  magnet,  cut  by  the  conductor,  C. 


FIG.  215. — Diagrammatic  illustration  of  unipolar  machine  with  two  high- 
speed collectors. 

Such  a  machine  is  called  a  unipolar  machine,  as  the  conductor 
during  its  rotation  traverses  the  same  polarity,  in  distinction  of 
bipolar  or  multipolar  machines,  in  which  the  conductor  during 
each  revolution  passes  two  or  many  poles.  A  more  correct  name 
is  homopolar  machine,  signifying  uniformity  of  polarity,  or- 
acyclic  machine,  signifying  absence  of  any  cyclic  change:  in  all 
other  electromagnetic  machines,  the  voltage  induced  in  a  con- 
ductor changes  cyclically,  and  the  voltage  in  each  turn  is  alter- 
nating, thus  having  a  frequency,  even  if  the  terminal  voltage 
and  current  at  the  commutator  are  continuous. 

450 


UNIPOLAR  MACHINES  451 

By  bringing  the  conductor,  C,  over  the  end  of  the  magnet  close 
to  the  shaft,  as  shown  in  Fig.  216,  the  peripheral  speed  of  motion 
of  brush,  Bz,  on  its  collector  ring  can  be  reduced.  However,  at 
least  one  brush,  BI,  in  Fig.  216,  must  bear  on  a  collector  ring 
(not  shown  in  Figs.  215  and  216)  at  full  conductor  speed,  because 
the  total  magnetic  flux  cut  by  the  conductor,  C,  must  pass  through 
this  collector  ring  on  which  B\  bears.  Thus  an  essential  char- 
acteristic of  the  unipolar  machine  is  collection  of  the  current  from 
the  periphery  of  the  revolving  conductor,  at  its  maximum  speed. 
It  is  the  unsolved  problem  of  satisfactory  current  collection  from 
high-speed  collector  rings,  at  speeds  of  two  or  more  miles  per 


FIG.  216. — Diagrammatic  illustration  of  unipolar  machine  with  one  high- 
speed collector. 

minute,   which  has  stood  in  the  way  of  the  commercial  intro- 
duction of  unipolar  machines. 

Electromagnetic  induction  is  due  to  the  relative  motion  of  con-' 
ductor  and  magnetic  field,  and  every  electromagnetic  device  is 
thus  reversible  with  regards  to  stationary  and  rotary  elements. 
However,  the  hope  of  eliminating  high-speed  collector  rings  in 
the  unipolar  machine,  by  having  the  conductor  standstill  and 
the  magnet  revolve,  is  a  fallacy:  in  Figs.  215  and  216,  the  con- 
ductor, C,  revolves,  and  the  magnet,  NS,  and  the  external  circuit, 
D,  stands  still.  The  mechanical  reversal  thus  would  be,  to  have 
the  conductor,  C,  stand  still,  and  the  magnet,  NS,  and  the  external 
circuit  revolve,  and  this  would  leave  high-speed  current  collection. 

Whether  the  magnet,  NS,  stands  still  or  revolves,  is  immaterial 
in  any  case,  and  the  question,  whether  the  lines  of  force  of  the 
magnet  are  stationary  or  revolve,  if  the  magnet  revolves  around 
its  axis,  is  meaningless.  If,  with  revolving  conductor,  C,  and 
stationary  external  circuit,  D,  the  lines  of  force  of  the  magnet 
are  assumed  as  stationary,  the  induction  is  in  C,  and  the  return 
circuit  in  D;  if  the  lines  of  force  are  assumed  as  revolving,  the 


452  ELECTRICAL  APPARATUS 

induction  is  in  D,  and  C  is  the  return,  but  the  voltage  in  the  cir- 
cuit, CD,  is  the  same.  If,  then,  C  and  D  both  stand  still,  either 
there  is  no  induction  in  either,  or,  assuming  the  lines  of  magnetic 
force  to  revolve,  equal  and  opposite  voltages  are  induced  in  C 
and  D,  and  the  voltage  in  circuit,  CD,  is  zero  just  the  same. 
However,  the  question  whether  the  lines  of  force  of  a  revolving 
magnet  rotate  or  not,  is  meaningless  for  this  reason:  the  lines  of 
force  are  a  pictorial  representation  of  the  magnetic  field  in  space. 
The  magnetic  field  at  any  point  is  characterized  by  an  intensity 
and  a  direction,  and  as  long  as  intensity  and  direction  at  any  point 
are  constant  or  stationary,  the  magnetic  field  is  constant  or  sta- 
tionary. This  is  the  case  in  Figs.  215  and  216,  regardless  whether 
the  magnet  revolves  around  its  axis  or  not,  and  the  rotation  of 
the  magnet  thus  has  no  effect  whatsoever  on  the  induction  phe- 
nomena. The  magnetic  field  is  stationary  at  any  point  of  space 
outside  of  the  magnet,  and  it  is  also  stationary  at  any  point  of 
space  inside  of  the  magnet,  even  if  the  magnet  revolves,  and  at 
the  same  time  it  is  stationary  also  with  regards  to  any  element 
of  the  revolving  magnet.  Using  then  the  pictorial  representation 
of  the  lines  of  magnetic  force,  we  can  assume  these  lines  of  force 
as  stationary  in  space,  or  as  revolving  with  the  rotating  magnet, 
whatever  best  suits  the  convenience  of  the  problem  at  hand:  but 
whichever  assumption  we  make,  makes  no  difference  on  the  solu- 
tion of  the  problem,  if  we  reason  correctly  from  the  assumption. 

248.  As  in  the  unipolar  machine  each  conductor  (correspond- 
ing to  a  half  turn  of  the  bipolar  or  multipolar  machine)  requires 
a  separate  high-speed  collector  ring,  many  attempts  have 
been  made  (and  are  still  being  made)  to  design  a  coil-wound 
unipolar  machine,  that  is,  a  machine  connecting  a  number  of 
peripheral  conductors  in  series,  without  going  through  collector 
rings.  This  is  an  impossibility,  and  unipolar  induction,  that  is, 
continues  induction  of  a  unidirectional  voltage,  is  possible  only 
in  an  open  conductor,  but  not  in  a  coil  or  turn,  as  the  voltage 
electromagnetically  induced  in  a  coil  or  turn  must  always  be  an 
alternating  voltage. 

The  fundamental  law  of  electromagnetic  induction  is,  that  the 
induced  voltage  is  proportional  to  the  rate  of  cutting  of  the  con- 
ductor through  the  lines  of  force  of  the  magnetic  field.  Applying 
this  to  a  closed  circuit  or  turn:  every  line  of  magnetic  force  cut 
by  a  turn  must  either  go  from  the  outside  to  the  inside,  or  from 
the  inside  to  the  outside  of  the  turn.  This  means:  the  voltage 


UNIPOLAR  MACHINES 


453 


induced  in  a  turn  is  proportional  (or  equal,  in  absolute  units)  to 
the  rate  of  change  of  the  number  of  lines  of  magnetic  force  en- 
closed by  the  turn,  and  a  decrease  of  the  lines  of  force  enclosed 
by  the  turn,  induces  a  voltage  opposite  to  that  induced  by  an 
increase.  As  the  number  of  lines  of  force  enclosed  by  a  turn  can 
not  perpetually  increase  (or  decrease),  it  follows,  that  a  voltage 
can  not  be  induced  perpetually  in  the  same  direction  in  a  turn. 
Every  increase  of  lines  of  force  enclosed  by  the  turn,  inducing 


FIG.    217. — Mechanical    an- 
alogy of  bipolar  induction. 


FIG.  218. — Mechanical  analogy  of 
unipolar  induction. 


a  voltage  in  it,  must  sometime  later  be  followed  by  an  equal 
decrease  of  the  lines  of  force  enclosed  by  the  turn,  which  induces 
an  equal  voltage  in  opposite  direction.  Thus,  averaged  over  a 
sufficiently  long  time,  the  total  voltage  induced  in  a  turn  must 
always  be  zero,  that  is,  the  voltage,  if  periodical,  must  be  alter- 
nating, regardless  how  the  electromagnetic  induction  takes  place, 
whether  the  turn  is  stationary  or  moving,  as  a  part  of  a  machine, 
transformer,  reactor  or  any  other  electromagnetic  induction 
device.  Thus  continuous-voltage  induction  in  a  closed  turn 
is  impossible,  and  the  coil- wound  unipolar  machine  thus  a 
fallacy.  Continuous  induction  in  the  unipolar  machine  is  pos- 
sible only  because  the  circuit  is  not  a  closed  one,  but  consists  of  a 
conductor  or  half  turn,  sliding  over  the  other  half  turn.  Mechan- 
ically the  relation  can  be  illustrated  by  Figs.  217  and  218.  If 
in  Fig.  217  the  carriage,  C,  moves  along  the  straight  track  of 
finite  length — a  closed  turn  of  finite  area — the  area,  A,  in  front  of 
C  decreases,  that  B  behind  the  carriage,  C,  increases,  but  this 
decrease  and  increase  can  not  go  on  indefinitely,  but  at  some  time 
C  reaches  the  end  of  the  track,  A  has  decreased  to  zero,  B  is  a 


454 


ELECTRICAL  APPARATUS 


FIG.  219. — Drum  type  of 
unipolar  machine  with  sta- 
tionary magnet  core,  section. 


maximum,  and  any  further  change  can  only  be  an  increase  of  A 
and  decrease  of  B,  by  a  motion  of  C  in  opposite  direction,  repre- 
senting induction  of  a  reverse  voltage.  On  the  endless  circular 

track,  Fig.  218,  however,  the  carriage, 
C,  can  continuously  move  in  the  same 
direction,  continuously  reduce  the 
area,  A,  in  front  and  increase  that  of 
B  behind  C,  corresponding  to  con- 
tinuous induction  in  the  same  direc- 
tion, in  the  unipolar  machine. 

249.  In  the  industrial  design  of  a 
unipolar  machine,  naturally  a  closed 
magnetic  circuit  would  be  used,  and 
the  form,  Fig.  216,  would  be  exe- 
cuted as  shown  in  length  section  in  Fig.  219.  N  is  the  same 
pole  as  in  Fig.  216,  but  the  magnetic  return  circuit  is  shown 
by  S,  concentrically  surrounding  N.  C  is  the  cylindrical  con- 
ductor, revolving  in  the  cylindrical  gap  be- 
tween N  and  S.  BI  and  B2  are  the  two  sets 
of  brushes  bearing  on  the  collector  rings  at 
the  end  of  the  conductor,  C,  and  F  is  the 
field  exciting  winding. 

The  construction,  Fig.  219,  has  the  me- 
chanical disadvantage  of  a  relatively  light 
structure,  C,  revolving  at  high  speed  between 
two  stationary  structures,  N  and  S.  As  it  is 
immaterial  whether  the  magnet  is  stationary 
or  revolving,  usually  the  inner  core,  N,  is  re- 
volved with  the  conductor,  as  shown  in 
Figs.  221  and  222.  This  shortens  the  gap 
between  N  and  S}  but  introduces  an  aux- 
iliary gap,  G.  Fig.  221  has  the  disadvantage 
of  a  magnetic  end  thrust,  and  thus  the  con- 
struction, Fig.  222,  is  generally  used,  or  its 
duplication,  shown  in  Fig.  223. 

The  disk  type  of  unipolar  machine,  shown 
in  section  in  Fig.  220,  has  been  frequently  proposed  in  former 
times,  but  is  economically  inferior  to  the  construction  of  Figs. 
221,  222  and  223.  The  limitation  of  the  unipolar  machine  is  the 
high  collector  speed.  In  Fig.  220,  the  average  conductor  speed 
is  less  than  the  collector  speed,  and  the  latter  thus  relatively 


FIG.  220. — Disc 
type  of  unipolar  ma- 
chine, section. 


UNIPOLAR  MACHINES 


455 


higher  than  in  Figs.  221  to  223,  where  it  equals  the  conductor 
speed. 

Higher  voltages  then  can  be  given  by  a  single  conductor,  are 

F     ?•  f     B. 


FIG.  221. — Drum  type  of  unipolar  FIG.  222. — Drum  type  of  unipolar 
machine  with  revolving  magnet  core  machine  with  revolving  magnet  core 
and  auxiliary  end  gap,  section.  and  auxiliary  cylinder  gap,  section. 

derived  in  the  unipolar  machine  by  connecting  a  number  of  con- 
ductors in  series.     In  this  case,  every  series  conductor  obviously 


FIG.  223. — Double  drum  type  of  unipolar  machine,  section. 

requires  a  separate  pair  of  collector  rings.     This  is  shown  in  Figs. 
224  and  225,  the  cross-section  and  length  section  of  the  rotor  of 


1  2  34 


4,3,2,1, 


ID 


FIG.  224. — Multi-    FIG.  225. — Multi-conductor  unipolar  machine, 
conductor  unipolar  length  section, 

machine,  cross-sec- 
tion. 

a  four-circuit  unipolar.     As  seen  in  Fig.  224,  the  cylindrical  con- 
ductor is  slotted  into  eight  sections,  and  Diametrically  opposite 


456  ELECTRICAL  APPARATUS 

sections,  1  and  1',  2  and  2',  3  and  3',  4  and  4',  are  connected  in 
multiple  (to  equalize  the  flux  distribution)  between -four  pairs  of 
collector  rings,  shown  in  Fig.  225  as  1  and  li,  2  and  2i,  3  and  3i, 
4  and  4i.  The  latter  are  connected  in  series.  This  machine, 
Figs.  224  and  225,  thus  could  also  be  used  as  a  three-wire  or 
five- wire  machine,  or  as  a  direct-current  converter,  by  bringing 
out  intermediary  connections,  from  the  collector  rings  2,  3,  4. 

250.  As  each  conductor  of  the  unipolar  machine  requires  a 
separate  pair  of  collector  rings,  with  a  reasonably  moderate 
number  of  collector  rings,  unipolar  machines  of  medium  capacity 
are  suited  for  low  voltages  only,  such  as  for  electrolytic  machines, 
and  have  been  built  for  this  purpose  to  a  limited  extent,  but  in 
general  it  has  been  found  more  economical  by  series  connection 
of  the  electrolytic  cells  to  permit  the  use  of  higher  voltages,  and 
then  employ  standard  machines. 

For  commercial  voltages,  250  or  600,  to  keep  the  number  of 
collector  rings  reasonably  moderate,  unipolar  machines  require 
very  large  magnetic  fluxes — that  is,  large  units  of  capacity — and 
very  high  peripheral  speeds.  The  latter  requirement  made  this 
machine  type  unsuitable  during  the  days  of  the  slow-speed  direct- 
connected  steam  engine,  but  when  the  high-speed  steam  turbine 
arrived,  the  study  of  the  design  of  high-powered  steam-turbine- 
driven  unipolars  was  undertaken,  and  a  number  of  such  machines 
built  and  installed. 

In  the  huge  turbo-alternators  of  today,  the  largest  loss  is  the 
core  loss:  hysteresis  and  eddies  in  the  iron,  which  often  is  more 
than  all  the  other  losses  together.  Theoretically,  the  unipolar 
machine  has  no  core  loss,  as  the  magnetic  flux  does  not  change 
anywhere,  and  solid  steel  thus  is  used  throughout — and  has  to 
be  used,  due  to  the  shape  of  the  magnetic  circuit.  However, 
with  the  enormous  magnetic  fluxes  of  these  machines,  in  solid 
iron,  the  least  variation  of  the  magnetic  circuit,  such  as  caused 
by  small  unequalities  of  the  air  gap,  by  the  reaction  of  the  arma- 
ture currents,  etc.,  causes  enormous  core  losses,  mostly  eddies, 
and  while  theoretically  the  unipolar  has  no  core  loss,  designing 
experience  has  shown,  that  it  is  a  very  difficult  problem  to  keep 
the  core  loss  in  such  machines  down  to  reasonable  values.  Fur- 
thermore, in  and  at  the  collector  rings,  the  magnetic  reaction  of 
the  armature  currents  is  alternating  or  pulsating.  Thus  in  Figs. 
224  and  225,  the  point  of  entrance  of  the  current  from  the  arma- 
ture conductors  into  the  collector  rings  revolves  with  the  rotation 


UNIPOLAR  MACHINES  457 

of  the  machine,  and  from  this  point  flows  through  the  collector 
ring,  distributing  between  the  next  brushes.  While  this  circular 
flow  of  current  in  the  collector  ring  represents  effectively  a  frac- 
tion of  a  turn  only,  with  thousands  of  amperes  of  current  it 
represents  thousands  of  ampere-turns  m.m.f .,  causing  high  losses, 
which  in  spite  of  careful  distribution  of  the  brushes  to  equalize 
the  current  flow  in  the  collector  rings,  can  not  be  entirely 
eliminated. 

251.  The  unipolar  machine  is  not  free  of  armature  reaction,  as 
often  believed.  The  current  in  all  the  armature  conductors 
(Fig.  224)  flows  in  the  same  direction,  and  thereby  produces  a 
circular  magnetization  in  the  magnetic  return  circuit,  S,  shown 
by  the  arrow  in  Fig.  224.  While  the  armature  conductor  mag- 
netically represents  one  turn  only,  in  the  large  machines  it  repre- 
sents many  thousand  ampere-turns.  As  an  instance,  assume  a 
peripheral  speed  of  a  steam-turbine-driven  unipolar  machine,  of 
12,000  ft.  per  minute,  at  1800  revolutions  per  minute.  This 
gives  an  armature  circumference  of  80  in.  At  J£  in.  thickness 
of  the  conductor,  and  2500  amp.  per 
square  inch,  this  gives  100,000  ampere- 
turns  m.m.f.  of  armature  reaction, 
which  probably  is  sufficient  to  magnetic- 
ally saturate  the  iron  in  the  pole  faces,  in 
the  direction  of  the  arrow  in  Fig.  224. 
At  the  greatly  lowered  permeability  at 
saturation,  with  constant  field  excita- 
tion the  voltage  of  the  machine  greatly 

drops,  or,  to  maintain  constant  voltage,       JTIG   226. Multi-con- 

a  considerable  increase  of  field  excita-  ductor  unipolar  machine 
. .  111-  -IT  with  compensating  pole 

tion    under    load     is     required.      Large   face  winding,  cross-section. 

unipolar    machines    thus   are  liable    to 

give  poor  voltage  regulation  and  to  require  high  compounding. 

To  overcome  the  circular  armature  reaction,  a  counter  m.m.f. 
may  be  arranged  in  the  pole  faces,  by  returning  the  current  of 
each  collector  ring  li,  2i,  3i,  4i,  of  Fig.  225,  to  the  collector  rings 
on  the  other  end  of  the  machine,  2,  3,  4  in  Fig.  225,  not  through 
an  external  circuit,  but  through  conductors  imbedded  in  the  pole 
face,  as  shown  in  Fig.  226  as  I',  2',  3',  4'. 

The  most  serious  problem  of  the  unipolar  machine,  however, 
is  that  of  the  high-speed  collector  rings,  and  this  has  not  yet  been 
solved.  Collecting  very  large  currents  by  numerous  collector 


458 


ELECTRICAL  APPARATUS 


rings  at  speeds  of  10,000  to  15,000  ft.  per  minute,  leads  to  high 
losses  and  correspondingly  low  machine  efficiency,  high  tempera- 
ture rise,  and  rapid  wear  of  the  brushes  and  collector  rings,  and 
this  has  probably  been  the  main  cause  of  abandoning  the  develop- 
ment of  the  unipolar  machine  for  steam-turbine  drive. 

A  contributing  cause  was  that,  when  the  unipolar  steam-tur- 
bine generator  was  being  developed,  the  days  of  the  huge  direct- 
current  generator  were  over,  and  its  place  had  been  taken  by 
turbo-alternator  and  converter,  and  the  unipolar  machine  offered 


FIG.  227. — Unipolar  motor  meter. 

no  advantage  in  reliability,  or  efficiency,  but  the  disadvantage 
of  lesser  flexibility,  as  it  requires  a  greater  concentration  of  direct- 
current  generation  in  one  place,  than  usually  needed. 

252.  The  unipolar  machine  may  be  used  as  motor  as  well 
as  generator,  and  has  found  some  application  as  motor  meter. 
The  general  principle  of  a  unipolar  meter  may  be  illustrated  by 
Fig.  227. 

The  meter  shaft,  A,  with  counter,  F,  is  pivoted  at  P,  and  carries 
the  brake  disk  and  conductor,  a  copper  or  aluminum  disk,  D,  be- 
tween the  two  poles,  N  and  S,  of  a  circular  magnet.  The  shaft,  A, 
dips  into  a  mercury  cup,  C,  which  is  insulated  and  contains  the 
one  terminal,  while  the  other  terminal  goes  to  a  circular  mercury 
trough,  G.  An  iron  pin,  B,  projects  from  the  disk,  D,  into  this 
mercury  trough  and  completes  the  circuit. 


CHAPTER  XXIII 
REVIEW 

253.  In  reviewing  the  numerous  types  of  apparatus,  methods 
of  construction  and  of  operation,  discussed  in  the  preceding, 
an  alphabetical  list  of  them  is  given  in  the  following,  comprising 
name,  definition,  principal  characteristics,  advantages  and  dis- 
advantages, and  the  paragraph  in  which  they  are  discussed. 

Alexanderson  High-frequency  Inductor  Alternator. — 159. 
Comprises  an  inductor  disk  of  very  many  teeth,  revolving  at  very 
high  speed  between  two  radial  armatures.  Used  for  producing 
very  high  frequencies,  from  20,000  to  200,000  cycles  per  second. 

Amortisseur. — Squirrel-cage  winding  in  the  pole  faces  of  the 
synchronous  machine,  proposed  by  Leblanc  to  oppose  the  hunt- 
ing tendency,  and  extensively  used. 

Amplifier. — 161.  An  apparatus  to  intensify  telephone  and 
radio  telephone  currents.  High-frequency  inductor  alternator 
excited  by  the  telephone  current,  usually  by  armature  reaction 
through  capacity.  The  generated  current  is  then  rectified,  be- 
fore transmission  in  long-distance  telephony,  after  transmission 
in  radio  telephony. 

Arc  Machines. — 138.  Constant-current  generators,  usually 
direct-current,  with  rectifying  commutators.  The  last  and  most 
extensively  used  arc  machines  were : 

Brush  Arc  Machine. — 141-144.  A  quarter-phase  constant- 
current  alternator  with  rectifying  commutators. 

Thomson-Houston  Arc  Machine. — 141-144.  A  three-phase 
F-connected  constant-current  alternator  with  rectifying  commu- 
tator. 

The  development  of  alternating-current  series  arc  lighting  by 
constant-current  transformers  greatly  reduced  the  importance 
of  the  arc  machine,  and  when  in  the  magnetite  lamp  arc 
lighting  returned  to  direct  current,  the  development  of  the 
mercury-arc  rectifier  superseded  the  arc  machine. 

Asynchronous  Motor. — Name  used  for  all  those  types  of 
alternating-current  (single-phase  or  polyphase)  motors  or  motor 
couples,  which  approach  a  definite  synchronous  speed  at  no-load, 
and  slip  below  this  speed  with  increasing  load. 

459 


460  ELECTRICAL  APPARATUS 

Brush  Arc  Machine. — (See  "Arc  Machines.") 

Compound  Alternator. — 138.  Alternator  with  rectifying  com- 
mutator, connected  in  series  to  the  armature,  either  con- 
ductively,  or  inductively  through  transformer,  and  exciting  a 
series  field  winding  by  the  rectified  current.  The  limitation  of 
the  power,  which  can  be  rectified,  and  the  need  of  readjusting  the 
brushes  with  a  change  of  the  inductivity  of  the  load,  has  made  such 
compounding  unsuitable  for  the  modern  high-power  alterna- 
tors. 

Condenser  Motor. — 77.  Single-phase  induction  motor  with 
condenser  in  tertiary  circuit  on  stator,  for  producing  starting 
torque  and  high  power-factor.  The  space  angle  between  pri- 
mary and  tertiary  stator  circuit  usually  is  45°  to  60°,  and  often  a 
three-phase  motor  is  used,  with  single-phase  supply  on  one  phase, 
and  condenser  on  a  second  phase.  With  the  small  amount  of 
capacity,  sufficient  for  power-factor  compensation,  usually  the 
starting  torque  is  small,  unless  a  starting  resistance  is  used,  but 
the  torque  efficiency  is  high. 

Concatenation. — III,  28.  Chain  connection,  tandem  connec- 
tion, cascade  connection.  Is  the  connection  o  the  secondary  of 
an  induction  machine  with  a  second  machine.  The  second 
machine  may  be: 

1.  An  Induction  Machine. — The  couple  then  is  asynchronous. 
Hereto  belong: 

The  induction  frequency  converter  or  general  alternating-current 
transformer,  XII,  103.  It  transforms  between  alternating-cur- 
rent systems  of  different  frequency,  and  has  over  the  induction- 
motor  generator  set  the  advantage  of  higher  efficiency  and  lesser 
capacity,  but  the  disadvantage  of  not  being  standard. 

The  concatenated  couple  of  induction  motors,  9,  28,  111.  It 
permits  multispeed  operation.  It  has  the  disadvantage  against 
the  multispeed  motor,  that  two  motors  are  required;  but  where 
two  or  more  motors  are  used,  as  in  induction-motor  railroading, 
it  has  the  advantage  of  greater  simplicity. 

The  internally  concatenated  motor  (Hunt  motor),  36.  It  is 
more  efficient  than  the  concatenated  couple  or  the  multispeed 
motor,  but  limited  in  design  to  certain  speeds  and  speed  ratios. 

2.  A  Synchronous  Machine. — The  couple  then  is  synchronous. 
Hereto  belong: 

The  synchronous  frequency  converter,  XII,  103.  It  has  a  defi- 
nite frequency  ratio,  while  that  of  the  induction  frequency  con- 


REVIEW  461 

verier  slightly  changes  with  the  load,  by  the  slip  of  the  induction 
machine. 

Induction  Motor  with  Low-frequency  Synchronous  Exciter. — 47. 
The  synchronous  exciter  in  this  case  is  of  small  capacity,  and 
gives  speed  control  and  power-factor  compensation. 

InductionGeneratorwithLow-frequency  Exciter. — 110, 121.  Syn- 
chronous induction  generator.  Stanley  induction  generator.  In 
this  case,  the  low-frequency  exciter  may  be  a  synchronous  or  a 
commutating  machine  or  any  other  source  of  low  frequency. 
The  phase  rotation  of  the  exciter  may  be  in  the  reverse  direc- 
tion of  the  main  machine,  or  in  the  same  direction.  In  the  first 
case,  the  couple  may  be  considered  as  a  frequency  converter 
driven  backward  at  many  times  synchronous  speed,  the  exciter 
is  motor,  and  the  generated  frequency  less  than  the  speed.  In 
the  case  of  the  same  phase  rotation  of  exciter  and  main  machine, 
the  generated  frequency  is  higher  than  the  speed,  and  the 
exciter  also  is  generator.  This  synchronous  induction  generator 
has  peculiar  regulation  characteristics,  as  the  armature  reaction 
of  non-inductive  load  is  absent. 

3.  A  Synchronous  Commutating  Machine. — 112.     The  couple 
is  synchronous,  and  called  motor  converter.     It  has  the  advantage 
of  lower  frequency  commutation,  and  permits  phase  control  by 
the  internal  reactance  of  the  induction  machine.     It  has  higher 
efficiency  and  smaller  size  than  a  motor-generator  set,  but  is 
larger  and  less  efficient  than  the  synchronous  converter,  and 
therefore  has  not  been  able  to  compete  with  the  latter. 

4.  A  direct-current  commutating  machine,  as  exciter,  41.     This 
converts  the  induction   machine  into   a  synchronous  machine 
(Danielson  motor).     A  good  induction  motor  gives  a  poor  syn- 
chronous motor,  but  a  bad  induction  motor,  of  very  low  power- 
factor,  gives  a  good  synchronous  motor,  of  good  power-factor, 
etc. 

5.  An    alternating-current    commutating    machine,  as  low-fre- 
quency exciter,  52.     The  couple  then  is  asynchronous.     This 
permits  a  wide  range  of  power-factor  and  speed  control  as  motor. 
As  generator  it  is  one  form  of  the  Stanley  induction  generator 
discussed  under  (2). 

6.  A   Condenser. — This   permits   power-factor   compensation, 
55,    and   speed    control,    11.     The   power- factor    compensation 
gives  good  values  with  very  bad  induction  motors,  of  low  power- 
factor,  but  is  uneconomical  with  good  motors.     Speed  control 


462  ELECTRICAL  APPARATUS 

usually  requires  excessive  amounts  of  capacity,  and  gives  rather 
poor  constants.  The  machine  is  asynchronous. 

Danielson  Motor. — 41.  An  induction  motor  converted  to  a 
synchronous  motor  by  direct-current  excitation.  (See  "  Con- 
catenation (4).") 

Deep-bar  Induction  Motor. — 7.  Induction  motor  with  deep 
and  narrow  rotor  bars.  At  the  low  frequency  near  synchronism, 
the  secondary  current  traverses  the  entire  rotor  conductor,  and 
the  secondary  resistance  thus  is  low.  At  high  slips,  as  in  start- 
ing, unequal  current  distribution  in  the  rotor  bars  concentrates 
the  current  in  the  top  of  the  bars,  thus  gives  a  greatly  increased 
effective  resistance,  and  thereby  higher  torque.  However,  the 
high  reactance  of  the  deep  bar  somewhat  impairs  the  power- 
factor.  The  effect  is  very  closely  the  same  as  in  the  double 
squirrel  cage.  (See  "Double  Squirrel-cage  Induction  Motor.") 

Double  Squirrel-cage  Induction  Motor. — II,  18.  Induction 
motor  having  a  high-resistance  low- reactance  squirrel  cage,  close 
to  the  rotor  surface,  and  a  low-resistance  high-reactance  squirrel 
cage,  embedded  in  the  core.  The  latter  gives  torque  at  good 
speed  regulation  near  synchronism,  but  carries  little  current  at 
lower  speeds,  due  to  its  high  reactance.  The  surface  squirrel 
cage  gives  high  torque  and  good  torque  efficiency  at  low  speeds 
and  standstill,  due  to  its  high  resistance,  but  little  torque  near 
synchronism.  The  combination  thus  gives  a  uniformly  high 
torque  over  a  wide  speed  range,  but  at  some  sacrifice  of  power- 
factor,  due  to  the  high  reactance  of  the  lower  squirrel  cage.  To 
get  close  speed  regulation  near  synchronism,  together  with  high 
torque  over  a  very  wide  speed  range,  for  instance,  down  to  full 
speed  in  reverse  direction  (motor  brake),  a  triple  squirrel  cage 
may  be  used,  one  high  resistance  low  reactance,  one  medium 
resistance  and  reactance,  and  one  very  low  resistance  and  high 
reactance  (24). 

Double  Synchronous  Machine. — 110,  119.  An  induction  ma- 
chine, in  which  the  rotor,  running  at  double  synchronism,  is 
connected  with  the  stator,  either  in  series  or  in  parallel,  but  with 
reverse  phase  rotation  of  the  rotor,  so  that  the  two  rotating  fields 
coincide  and  drop  into  step  at  double  synchronism.  The  machine 
requires  a  supply  of  lagging  current  for  excitation,  just  like  any 
induction  machine.  It  may  be  used  as  synchronous  induction 
generator,  or  as  synchronous  motor.  As  generator,  the  armature 
reaction  neutralizes  at  non-inductive,  but  not  at  inductive  load, 


REVIEW  463 

and  thus  gives  peculiar  regulation  characteristics,  similar  as  the 
Stanley  induction  generator.  It  has  been  proposed  for  steam- 
turbine  alternators,  as  it  would  permit  higher  turbine  speed 
(3000  revolutions  at  25  cycles)  but  has  not  yet  been  used.  As 
motor  it  has  the  disadvantage  that  it  is  not  self-starting. 

Eickemeyer  Inductively  Compensated  Single-phase  Series 
Motor. — 193.  Single-phase  commutating  machine  with  series 
field  and  inductive  compensating  winding. 

Eickemeyer  Inductor  Alternator. — 160.  Inductor  alternator 
with  field  coils  parallel  to  shaft,  so  that  the  magnetic  flux  disposi- 
tion is  that  of  a  bipolar  or  multipolar  machine,  in  which  the 
multitooth  inductor  takes  the  place  of  the  armature  of  the  stand- 
ard machine.  Voltage  induction  then  takes  place  in  armature 
coils  in  the  pole  faces,  and  the  magnetic  flux  in  the  inductor  re- 
verses, with  a  frequency  much  lower  than  that  of  the  induced 
voltage.  This  type  of  inductor  machine  is  specially  adopted  for 
moderately  high  frequencies,  300  to  2000  cycles,  and  used  in  in- 
ductor alternators  and  inductor  converters.  In  the  latter,  the  in- 
ductor carries  a  low-frequency  closed  circuit  armature  winding 
connected  to  a  commutator  to  receive  direct  current  as  motor. 

Eickemeyer  Rotary  Terminal  Induction  Motor. — XI,  101. 
Single-phase  induction  motor  with  closed  circuit  primary  winding 
connected  to  commutator.  The  brushes  leading  the  supply  cur- 
rent into  the  commutator  stand  still  at  full  speed,  but  revolve 
at  lower  speeds  and  in  starting.  This  machine  can  give  full  maxi- 
mum torque  at  any  speed  down  to  standstill,  depending  on  the 
speed  of  the  brushes,  but  its  disadvantage  is  sparking  at  the  com- 
mutator, which  requires  special  consideration. 

Frequency  Converter  or  General  Alternating-current  Trans- 
former.— XII,  103.  Transforms  a  polyphase  system  into  another 
polyphase  system  of  different  frequency  and  where  desired  of  differ 
ent  voltage  and  different  number  of  phases.  Consists  of  an  induc- 
tion machine  concatenated  to  a  second  machine,  which  may  be 
an  induction  machine  or  a  synchronous  machine,  thus  giving  the 
induction  frequency  converter  and  the  synchronous  frequency  con- 
verter. (See  "  Concatenation/')  In  the  synchronous  frequency 
converter  the  frequency  ratio  is  rigidly  constant,  in  the  induction 
frequency  converter  it  varies  slightly  with  the  load,  by  the  slip 
of  the  induction  machine.  When  increasing  the  frequency,  the 
second  machine  is  motor,  when  decreasing  the  frequency,  it  is 
generator.  Above  synchronism,  both  machines  are  generators 


464  ELECTRICAL  APPARATUS 

and  the  machine  thus  a  synchronous  induction  generator.  In 
concatenation,  the  first  machine  always  acts  as  frequency  con- 
verter. The  frequency  converter  has  the  advantage  of  lesser 
machine  capacity  than  the  motor  generator,  but  the  disadvantage 
of  not  being  standard  yet. 

Heyland  Motor. — 59,  210.  Squirrel-cage  induction  motor  with 
commutator  for  power-factor  compensation. 

Hunt  Motor. — 36.  .  Internally  concatenated  induction  motor. 
(See  " Concatenation  (1).") 

Hysteresis  Motor. — X,  98.  Motor  with  polyphase  stator  and 
laminated  rotor  of  uniform  reluctance  in  all  directions,  without 
winding.  Gives  constant  torque  at  all  speeds,  by  the  hysteresis 
of  the  rotor,  as  motor  below  and  as  generator  above  synchronism, 
while  at  synchronism  it  may  be  either.  Poor  power-factor  and 
small  output  make  it  feasible  only  in  very  small  sizes,  such  as 
motor  meters. 

Inductor  Machines. — XVII,  156.  Synchronous  machine,  gen- 
erator or  motor,  in  which  field  and  armature  coils  stand  still  and 
the  magnetic  field  flux  is  constant,  and  the  voltage  is  induced  by 
changing  the  flux  path,  that  is,  admitting  and  withdrawing  the 
flux  from  the  armature  coils  by  means  of  a  revolving  inductor. 
The  inducing  flux  in  the  armature  coils  thus  does  not  alternate, 
but  pulsates  without  reversal.  For  standard  frequencies  the 
inductor  machine  is  less  economical  and  little  used,  but  it  offers 
great  constructive  advantages  at  high  frequencies  and  is  the  only 
feasible  type  at  extremely  high  frequencies.  Excited  by  alter- 
nating currents,  the  inductor  machine  may  be  used  as  amplifier 
(see  " Amplifier");  excited  by  polyphase  currents,  it  is  an  induc- 
tion inductor  frequency  converter,  162;  with  a  direct-current  wind- 
ing on  the  inductor,  it  is  a  direct-current  high-frequency  converter. 
(See  "Eickemeyer  Inductor  Alternator.") 

Leading  current,  power-factor  compensation  and  phase  control 
can  be  produced  by: 

Condenser. 

Polarization  cell. 

Overexcited  synchronous  motor  or  synchronous  converter. 

Induction  machine  concatenated  to  condenser,  to  synchronous 
motor  or  to  low-frequency  commutating  machine. 

Alternating-current  commutating  machine  with  lagging  field 
excitation. 

Leblanc's  Panchahuteur. — 145.     Synchronous  rectifier  of  many 


REVIEW  465 

phases,  fed  by  polyphase  transformer  increasing  the  number  of 
phases,  and  driven  by  a  synchronous  motor  having  as  many  cir- 
cuits as  the  rectifier  has  phases,  each  synchronous  motor  circuit 
being  connected  in  shunt  to  the  corresponding  rectifier  phase  to 
bye  pass  the  differential  current  and  thereby  reduce  inductive 
sparking.  Can  rectify  materially  more  power  than  the  standard 
rectifier,  but  is  inferior  to  the  converter. 

Magneto  Commutation. — 163.  Apparatus  in  which  the  induc- 
tion is  varied,  with  stationary  inducing  (exciting)  and  induced 
coils,  by  shifting  or  reversing  the  magnetic  flux  path  by  means 
of  a  movable  part  of  the  magnetic  circuit,  the  inductor.  Applied 
to  stationary  induction  apparatus,  as  voltage  regulators,  and  to 
synchronous  machines,  as  inductor  alternator. 

Monocyclic. — 127.  A  system  of  polyphase  voltages  with  essen- 
tially single-phase  flow  of  power.  A  system  of  polyphase  vol- 
tages, in  which  one  phase  regulates  for  constant  voltage,  that  is, 
a  voltage  which  does  not  materially  drop  within  the  range  of 
power  considered,  while  the  voltage  in  quadrature  phase  thereto 
is  of  limited  power,  that  is,  rapidly  drops  with  increase  of  load. 
Monocyclic  systems,  as  the  square  or  the  triangle,  are  derived 
from  single-phase  supply  by  limited  energy  storage  in  inductance 
or  capacity,  and  used  in  those  cases,  as  single-phase  induction 
motor  starting,  where  the  use  of  a  phase  converter  would  be 
uneconomical. 

Motor  Converter. — 112.  An  induction  machine  concatenated 
with  a  synchronous  commutating  machine.  (See  "  Concatenation 
(3).")  The  latter  thus  receives  part  of  the  power  mechanically, 
part  electrically,  at  lower  frequency,  and  thereby  offers  the  ad- 
vantages incident  to  a  lower  frequency  in  a  commutating  machine. 
It  permits  phase  control  by  the  internal  reactance  of  the  induc- 
tion machine.  Smaller  than  a  motor-generator  set,  but  larger 
than  a  synchronous  converter,  and  the  latter  therefore  preferable 
where  it  can  be  used. 

Multiple  Squirrel-cage  Induction  Motor. — (See  "  Double 
Squirrel-cage  Induction  Motor.") 

Multispeed  Induction  Motor. — 14.  Polyphase  Induction 
Motor  with  the  primary  windings  arranged  so  that  by  the  opera- 
tion of  a  switch,  the  number  of  poles  of  the  motor,  and  thereby 
its  speed  can  be  changed.  It  is  the  most  convenient  method  of 
producing  several  economical  speeds  in  an  induction  motor,  and 
therefore  is  extensively  used.  At  the  lower  speed,  the  power- 
factor  necessarily  is  lower. 

30 


466  ELECTRICAL  APPARATUS 

Permutator. — 146.  Machine  to  convert  polyphase  alternating 
to  direct  current,  consisting  of  a  stationary  polyphase  trans- 
former with  many  secondary  phases  connected  to  a  stationary 
commutator,  with  a  set  of  revolving  brushes  driven  by  a  syn- 
chronous motor.  Thus  essentially  a  synchronous  converter 
with  stationary  armature  and  revolving  field,  but  with  two 
armature  windings,  primary  and  secondary.  The  foremost 
objection  is  the  use  of  revolving  brushes,  which  do  not  permit 
individual  observation  and  adjustment  during  operation,  and 
thus  are  liable  to  sparking. 

Phase  Balancer. — 134.  An  apparatus  producing  a  polyphase 
system  of  opposite  phase  rotation  for  insertion  in  series  to  a 
polyphase  system,  to  restore  the  voltage  balance  disturbed  by  a 
single-phase  load.  It  may  be: 

A  stationary  induction-phase  balancer,  consisting  of  an  induc- 
tion regulator  with  reversed  phase  rotation  of  the  series  winding. 

A  synchronous-phase  balancer,  consisting  of  a  synchronous 
machine  of  reversed  phase  rotation,  having  two  sets  of  field  wind- 
ings in  quadrature.  By  varying,  or  reversing  the  excitation  of  the 
latter,  any  phase  relation  of  the  balancer  voltage  with  those  of 
the  main  polyphase  system  can  be  produced.  The  synchronous 
phase  balancer  is  mainly  used,  connected  into  the  neutral  of  a 
synchronous  phase  converter,  to  control  the  latter  so  as  to  make 
the  latter  balance  the  load  and  voltage  of  a  polyphase  system 
with  considerable  single-phase  load,  such  as  that  of  a  single- 
phase  railway  system. 

Polyphase  Commutator  Motor. — Such  motors  may  be  shunt, 
181,  or  series  type,  187,  for  multispeed,  adjustable-speed  and 
varying-speed  service.  In  commutation,  they  tend  to  be  inferior 
to  single-phase  commutator  motors,  as  their  rotating  field  does 
not  leave  any  neutral  direction,  in  which  a  commutating  field 
could  be  produced,  such  as  is  used  in  single-phase  commutator 
motors.  Therefore,  polyphase  commutator  motors  have  been 
built  with  separate  phases  and  neutral  spaces  between  the  phases, 
for  commutating  fields :  Scherbius  motor. 

Reaction  Machines. — XVI,  147.  Synchronous  machine,  motor 
or  generator,  in  which  the  voltage  is  induced  by  pulsation  of  the 
magnetic  reluctance,  that  is,  by  make  and  break  of  the  magnetic 
circuit.  It  thus  differs  from  the  inductor  machine,  in  that  in 
the  latter' the  total  field  flux  is  constant,  but  is  shifted  with  re- 
gards to  the  armature  coils,  while  in  the  reaction  machine  the 


REVIEW  467 

total  field  flux  pulsates.  The  reaction  machine  has  low  output 
and  low  power-factor,  but  the  type  is  useful  in  small  synchronous 
motors,  due  to  the  simplicity  resulting  from  the  absence  of  direct- 
current  field  excitation. 

Rectifiers. — XV,  138.  Apparatus  to  convert  alternating  into 
direct  current  by  synchronously  changing  connections.  Rec- 
tification may  occur  either  by  synchronously  reversing  connec- 
tions between  alternating-current  and  direct-current  circuit: 
reversing  rectifier,  or  by  alternately  making  contact  between  the 
direct-current  circuit  and  the  alternating-current  circuit,  when 
the  latter  is  of  the  right  direction,  and  opening  contact,  when 
of  the  reverse  direction:  contact-making  rectifier.  Mechanical 
rectifiers  may  be  of  either  type.  Arc  rectifiers,  such  as  the  mer- 
cury-arc rectifier,  which  use  the  unidirectional  conduction  of  the 
arc,  necessarily  are  contact-making  rectifiers. 

Full-wave  rectifiers  are  those  in  which  the  direct-current  cir- 
cuit receives  both  half  waves  of  alternating  current;  half -wave 
rectifiers  those  in  which  only  alternate  half  waves  are  rectified, 
the  intermediate  or  reverse  half  waves  suppressed.  The  latter 
type  is  permissible  only  in  small  sizes,  as  the  interrupted  pul- 
sating current  traverses  both  circuits,  and  produces  in  the  alter- 
nating-current circuit  a  unidirectional  magnetization,  which 
may  give  excessive  losses  and  heating  in  induction  apparatus. 
The  foremost  objection  to  the  mechanical  rectifier  is,  that  the 
power  which  can  be  rectified  without  injurious  inductive  spark- 
ing, is  limited,  especially  in  single-phase  rectifiers,  but  for  small 
amounts  of  power,  as  for  battery  charging  and  constant-current 
arc  lighting  they  are  useful.  However,  even  there  the  arc  recti- 
fier is  usually  preferable.  The  brush  arc  machine  and  the 
Thomson  Houston  arc  machine  were  polyphase  alternators  with 
rectifying  commutators. 

Regulating  Pole  Converter. — Variable-ratio  converter.  Split- 
pole  converter,  XXI,  230.  A  synchronous  converter,  in  which 
the  ratio  between  direct-current  voltage  and  alternating-current 
voltage  can  be  varied  at  will,  over  a  considerable  range,  by  shift- 
ing the  direction  of  the  resultant  magnetic  field  flux  so  that  the 
voltage  between  the  commutator  brushes  is  less  than  maximum 
alternating-current  voltage,  and  by  changing,  at  constant  im- 
pressed effective  alternating  voltage,  the  maximum  alternating- 
current  voltage  and  with  it  the  direct-current  voltage,  by  the 
superposition  of  a  third  harmonic  produced  in  the  converter  in 


468  ELECTRICAL  APPARATUS 

such  a  manner,  that  this  harmonic  exists  only  in  the  local  con- 
verter circuit.  This  is  done  by  separating  the  field  pole  into 
two  parts,  a  larger  main  pole,  which  has  constant  excitation, 
and  a  smaller  regulating  pole,  in  which  the  excitation  is  varied 
and  reversed.  A  resultant  armature  reaction  exists  in  the 
regulating  pole  converter,  proportional  to  the  deviation  of  the 
voltage  ratio  from  standard,  and  requires  the  use  of  a  series  field. 
Regulating  pole  converters  are  extensively  used  for  adjustable 
voltage  service,  as  direct-current  distribution,  storage-battery 
charging,  etc.,  due  to  their  simplicity  and  wide  voltage  range  at 
practically  unity  power-factor,  while  for  automatic  voltage 
control  under  fluctuating  load,  as  railway  service,  phase  control 
of  the  standard  converter  is  usually  preferred. 

Repulsion  Generator. — 217.  Repulsion  motor  operated  as 
generator. 

Repulsion  Motor. — 194,  208,  214.  Single-phase  commutator 
motor  in  which  the  armature  is  short-circuited  and  energized 
by  induction  from  a  stationary  conpensating  winding  as  primary. 
Usually  of  varying  speed  or  series  characteristic.  Gives  better 
commutation  than  the  series  motor  at  moderate  speeds. 

Rotary  Terminal  Single-phase  Induction  Motor. — XI,  101. 
(See  "Eickemeyer  Rotary  Terminal  Induction  Motor.") 

Shading  Coil. — 73.  A  short-circuited  turn  surrounding  a  part 
of  the  pole  face  of  a  single-phase  induction  motor  with  definite 
poles,  for  the  purpose  of  giving  a  phase  displacement  of  the 
flux,  and  thereby  a  starting  torque.  It  is  the  simplest  and  cheap- 
est single-phase  motor-starting  device,  but  gives  only  low  start- 
ing torque  and  low  torque  efficiency,  thus  is  not  well  suited  for 
larger  motors.  It  thus  is  very  extensively  used  in  small  motors, 
almost  exclusively  in  alternating-current  fan  motors. 

Single-phase  Commutator  Motor. — XX,  189.  Commutator 
motor  with  alternating-current  field  excitation,  and  such  modi- 
fications of  design,  as  result  therefrom.  That  is,  lamination  of 
the  magnetic  structure,  high  ratio  of  armature  reaction  to  field 
excitation,  and  compensation  for  armature  reaction  and  self- 
induction,  etc.  Such  motor  thus  comprises  three  circuits:  the 
armature  circuit,  the  field  circuit,  and  the  compensating  circuit 
in  quadrature,  on  the  stator,  to  the  field  circuit.  These  cir- 
cuits may  be  energized  by  conduction,  from  the  main  current, 
or  by  induction,  as  secondaries  with  the  main  current  as  pri- 
mary. If  the  armature  receives  the  main  current,  the  motor  is 


REVIEW  469 

a  series  or  shunt  motor;  if  it  is  closed  upon  itself,  directly  or 
through  another  circuit,  the  motor  is  called  a  repulsion  motor. 
A  combination  of  both  gives  the  series  repulsion  motor. 

Single-phase  commutator  motors  of  series  characteristic  are 
used  for  alternating-current  railroading,  of  shunt  characteristic 
as  stationary  motors,  as  for  instance  the  induction  repulsion 
motor,  either  as  constant-speed  high-starting-torque  motors,  or 
as  adjustable-speed  motors. 

Lagging  the  field  magnetism,  as  by  shunted  resistance,  pro- 
duces a  lead  of  the  armature  current.  This  can  be  used  for 
power-factor  compensation,  and  single-phase  commutator  motors 
thereby  built  with  very  high  power-factors.  Or  the  machine, 
with  lagging  quadrature  field  excitation,  can  be  used  as  effective 
capacity.  The  single-phase  commutator  motor  is  the  only  type 
which,  with  series  field  excitation,  gives  a  varying-speed  motor 
of  series-motor  characteristics,  and  with  shunt  excitation  or  its 
equivalent,  give  speed  variation  and  adjustment  like  that  of  the 
direct-current  motor  with  field  control,  and  is  therefore  exten- 
sively used.  Its  disadvantage,  however,  is  the  difficulty  and 
limitation  in  design,  resulting  from  the  e.m.f .  induced  in  the  short- 
circuited  coils  under  the  brush,  by  the  alternation  of  the  main 
field,  which  tends  toward  sparking  at  the  commutator. 

Single-phase  Generation. — 135. 

Speed  Control  of  Polyphase  Induction  Motor.— 

By  resistance  in  the  secondary,  8.  Gives  a  speed  varying 
with  the  load. 

By  pyro-electric  resistance  in  the  secondary,  10.  Gives  good 
speed  regulation  at  any  speed,  but  such  pyro-electric  conductors 
tend  toward  instability. 

By  condenser  in  the  secondary,  11.  Gives  good  speed  regula- 
tion, but  rather  poor  power-factor,  and  usually  requires  an  un- 
economically  large  amount  of  capacity. 

By  commutator,  58.  Gives  good  speed  regulation  and  per- 
mits power-factor  control,  but  has  the  disadvantage  and  com- 
plication of  an  alternating-current  commutator. 

By  concatenation  with  a  low-frequency  commutating  machine 
as  exciter,  52.  Has  the  disadvantage  of  complication. 

Stanley  Induction  Generator. — 117.  Induction  machine  with 
low-frequency  exciter.  (See  " Concatenation  (2).") 

Stanley  Inductor  Alternator. — 159.  Inductor  machine  with 
two  armatures  and  inductors,  and  a  concentric  field  coil  between 
the  same.  (See  " Inductor  Machine.") 


470  ELECTRICAL  APPARATUS 

Starting  Devices. — Polyphase  induction  motor: 

Resistance  of  high  temperature  coefficient,  2.  Gives  good  torque 
curve  at  low  speed  and  good  regulation  at  speed,  but  requires 
high  temperature  in  the  resistance. 

Hysteresis  device,  4.  Gives  good  speed  regulation  and  good 
torque  at  low  speed  and  in  starting,  but  somewhat  impairs  the 
power-factor. 

Eddy-current  device,  5;  double  and  triple  squirrel-cage,  18,  20, 
24;  and  deep-bar  rotor,  7.  Give  good  speed  regulation  combined 
with  good  torque  at  low  speed  and  in  starting,  but  somewhat 
impairs  the  power-factor.  (See  "  Double  Squirrel-cage  Induction 
Motor"  and  " Deep-bar  Induction  Motor.") 

Single-phase  induction  motor: 

Phase-splitting  devices,  67.  Resistance  in  one  phase,  68.  In- 
ductive devices,  72.  Shading  coil,  73.  (See  "  Shading  Coil.") 
Monocyclic  devices,  76.  Resistance-reactance  device  or  mono- 
cyclic  triangle.  Condenser  motor,  77.  (See  "  Condenser  Motor.") 

Repulsion-motor  starting. 

Series-motor  starting. 

Synchronous-induction  Generator. — XIII,  113.  Induction 
machine,  in  which  the  secondary  is  connected  so  as  to  fix  a  definite 
speed.  This  may  be  done: 

1.  By  connecting  the  secondary,  in  reverse  phase  rotation,  in 
shunt  or  in  series  to  the  primary:  double  synchronous  generator. 
(See  "Double  Synchronous  Machine.") 

2.  By   connecting   the   secondary   in   shunt   to   the   primary 
through  a  commutator.     In  this  case,  the  resultant  frequency  is 
fixed  by  speed  and  ratio  of  primary  to  secondary  turns. 

3.  By  connecting  the  secondary  to  a  source  of  constant  low 
frequency:  Stanley  induction  generator.     In  this  case,  the  low- 
frequency  phase  rotation  impressed  upon  the  secondary  may  be 
in  the  same  or  in  opposite  direction  to  the  speed.     (See  "Con- 
catenation (2).") 

Synchronous-induction  Motor. — IX,  97.  An  induction  motor 
with  single-phase  secondary.  Tends  to  drop  into  step  as  syn- 
chronous motor,  and  then  becomes  generator  when  driven  by 
power.  Its  low  power-factor  makes  it  unsuitable  except  for 
small  sizes,  where  the  simplicity  due  to  the  absence  of  direct- 
current  excitation  may  make  it  convenient  as  self-starting  syn- 
chronous motor.  As  reaction  machine,  150. 

Thomson -Houston  Arc  Machine. — 141-144.     Three-phase  Y- 


REVIEW  471 

connected  constant-current  alternator  with  rectifying  commu- 
tator. 

Thomson  Repulsion  Motor. — 193.  Single-phase  compensated 
commutating  machine  with  armature  energized  by  secondary 
current,  and  field  coil  and  compensating  coil  combined  in  one  coil. 

Unipolar  Machines. — Unipolar  or  acyclic  machine,  XXII,  247. 
Machine  in  which  a  continuous  voltage  is  induced  by  the  rotation 
of  a  conductor  through  a  constant  and  uniform  magnetic  field. 
Such  machines  must  have  as  many  pairs  of  collector  rings  as  there 
are  conductors,  and  the  main  magnetic  flux  of  the  machine  must 
pass  through  the  collector  rings,  hence  current  collection  occurs 
from  high-speed  collector  rings.  Coil  windings  are  impossible 
in  unipolar  machines.  Such  machines  either  are  of  low  yoltage, 
or  of  large  size  and  high  speed,  thus  had  no  application  before 
the  development  of  the  high-speed  steam  turbine,  and  now  three- 
phase  generation  with  conversion  by  synchronous  converter  has 
eliminated  the  demand  for  very  large  direct-current  generating 
units.  The  foremost  disadvantage  is  the  high-speed  current 
collection,  which  is  still  unsolved,  and  the  liability  to  excessive 
losses  by  eddy  currents  due  to  any  asymmetry  of  the  magnetic 
field. 

Winter-Eichbery-Latour  Motor. — 194.  Single-phase  compen- 
sated series-type  motor  with  armature  excitation,  that  is,  the 
exciting  current,  instead  of  through  the  field,  passes  through  the 
armature  by  a  set  of  auxiliary  brushes  in  quadrature  with  the 
main  brushes.  Its  advantage  is  the  higher  power-factor,  due  to 
the  elimination  of  the  field  inductance,  but  its  disadvantage  the 
complication  of  an  additional  set  of  alternating-current  commu- 
tator brushes. 


CHAPTER  XXIV 
CONCLUSION 

254.  Numerous  apparatus,  structural  features  and  principles 
have  been  invented  and  more  or  less  developed,  but  have  found 
a  limited  industrial  application  only,  or  are  not  used  at  all,  be- 
cause there  is  no  industrial  demand  for  them.  Nevertheless  a 
knowledge  of  these  apparatus  is  of  great  importance  to  the  elec- 
trical engineer.  They  may  be  considered  as  filling  the  storehouse 
of  electrical  engineering,  waiting  until  they  are  needed.  Very 
often,  in  the  development  of  the  industry,  a  demand  arises  for 
certain  types  of  apparatus,  which  have  been  known  for  many 
years,  but  not  used,  because  they  offered  no  material  advan- 
tage, until  with  the  change  of  the  industrial  conditions  their 
use  became  very  advantageous  and  this  led  to  their  extensive 
application. 

Thus  for  instance  the  commutating  pole  ("interpole")  in 
direct-current  machines  has  been  known  since  very  many  years, 
has  been  discussed  and  recommended,  but  used  very  little,  in 
short  was  of  practically  no  industrial  importance,  while  now 
practically  all  larger  direct-current  machines  and  synchronous 
converters  use  commutating  poles.  For  many  years,  with  the 
types  of  direct-current  machines  in  use,  the  advantage  of  the 
commutating  pole  did  not  appear  sufficient  to  compensate  for 
the  disadvantage  of  the  complication  and  resultant  increase  of 
size  and  cost.  But  when  with  the  general  introduction  of  the 
steam-turbine  high-speed  machinery  became  popular,  and  higher- 
speed  designs  were  introduced  in  direct-current  machinery  also, 
with  correspondingly  higher  armature  reaction  and  greater  need 
of  commutation  control,  the  use  of  the  commutating  pole  became 
of  material  advantage  in  reducing  size  and  cost  of  apparatus, 
and  its  general  introduction  followed. 

Similarly  we  have  seen  the  three-phase  transformer  find  gen- 
eral introduction,  after  it  had  been  unused  for  many  years;  so 
also  the  alternating-current  commutator  motor,  etc. 

Thus  for  a  progressive  engineer,  it  is  dangerous  not  to  be  famil- 
iar with  the  characteristics  and  possibilities  of  the  known  but 

472 


CONCLUSION  473 

unused  types  of  apparatus,  since  at  any  time  circumstances  may 
arise  which  lead  to  their  extensive  introduction. 

255.  With  many  of  these  known  but  unused  or  little  used  ap- 
paratus, we  can  see  and  anticipate  the  industrial  condition  which 
will  make  their  use  economical  or  even  necessary,  and  so  lead  to 
their  general  introduction. 

Thus,  for  instance,  the  induction  generator  is  hardly  used  at  all 
today.  However,  we  are  only  in  the  beginning  of  the  water- 
power  development,  and  thus  far  have  considered  only  the  largest 
and  most  concentrated  powers,  and  for  these,  as  best  adapted, 
has  been  developed  a  certain  type  of  generating  station,  compris- 
ing synchronous  generators,  with  direct-current  exciting  circuits, 
switches,  circuit-breakers,  transformers  and  protective  devices, 
etc.,  and  requiring  continuous  attendance  of  expert  operating 
engineers.  This  type  of  generating  station  is  feasible  only  with 
large  water  powers.  As  soon,  however,  as  the  large  water  powers 
will  be  developed,  the  industry  will  be  forced  to  proceed  to  the 
development  of  the  numerous  scattered  small  powers.  That  is, 
the  problem  will  be,  to  collect  from  a  large  number  of  small 
water  powers  the  power  into  one  large  electric  system,  similar 
as  now  we  distribute  the  power  of  one  large  system  into  numer- 
ous small  consumption  places. 

The  new  condition,  of  collecting  numerous  small  powers — 
from  a  few  kilowatts  to  a  few  hundred  kilowatts — into  one  sys- 
tem, will  require  the  development  of  an  entirely  different  type  of 
generating  station:  induction  generators  driven  by  small  and 
cheap  waterwheels,  at  low  voltage,  and  permanently  connected 
through  step-up  transformers  to  a  collecting  line,  which  is  con- 
trolled from  some  central  synchronous  station.  A  cheap  hy- 
draulic development,  no  regulation  of  waterwheel  speed  or  gen- 
erator voltage,  no  attendance  in  the  station  beyond  an  occasional 
inspection,  in  short  an  automatically  operating  induction  gen- 
erator station  controlled  from  the  central  receiving  station. 

In  many  cases,  we  can  not  anticipate  what  application  an 
unused  type  of  apparatus  may  find,  and  when  its  use  may  be 
economically  demanded,  or  we  can  only  in  general  realize,  that 
with  the  increasing  use  of  electric  power,  and  with  the  intro- 
duction of  electricity  as  the  general  energy  supply  of  modern 
civilization,  the  operating  requirements  will  become  more  diver- 
sified, and  where  today  one  single  type  of  machine  suffices — as 
the  squirrel-cage  induction  motor — various  modifications  thereof 


474  ELECTRICAL  APPARATUS 

will  become  necessary,  to  suit  the  conditions  of  service,  such  as 
the  double  squirrel-cage  induction  motor  in  ship  propulsion  and 
similar  uses,  the  various  types  of  concatenation  of  induction 
machines  with  synchronous  and  commutating  machines,  etc. 

256.  In  general,   a  new  design  or  new  type  of  machine  or 
apparatus  has  economically  no  right  of  existence,  if  it  is  only 
just  as  good  as  the  existing  one. 

A  new  type,  which  offers  only  a  slight  advantage  in  efficiency, 
size,  cost  of  production  or  operation,  etc.,  over  the  existing  type, 
is  economically  preferable  only,  if  it  can  entirely  supersede  the 
existing  type;  but  if  its  advantage  is  limited  to  certain  applica- 
tions, very  often,  even  usually,  the  new  type  is  economically 
inferior,  since  the  disadvantage  of  producing  and  operating  two 
different  types  of  apparatus  may  be  greater  than  the  advantage 
of  the  new  type.  Thus  a  standard  type  is  economically  superior 
and  preferable  to  a  special  one,  even  if  the  latter  has  some  small 
superiority,  unless,  and  until,  the  industry  has  extended  so  far, 
that  both  types  can  find  such  extensive  application  as  to  justify 
the  existence  of  two  standard  types.  This,  for  instance,  was  the 
reason  which  retarded  the  introduction  of  the  three-phase  trans- 
former: its  advantage  was  not  sufficient  to  justify  the  dupli- 
cation of  standards,  until  three-phase  systems  had  become  very 
numerous  and  widespread. 

In  other  words,  the  advantage  offered  by  a  new  type  of  appara- 
tus over  existing  standard  types,  must  be  very  material,  to 
economically  justify  its  industrial  development. 

The  error  most  frequently  made  in  modern  engineering  is  not 
the  undue  adherence  to  standards,  but  is  the  reverse.  The 
undue  preference  of  special  apparatus,  sizes,  methods,  etc., 
where  standards  would  be  almost  as  good  in  their  characteristics, 
and  therefore  would  be  economically  preferable.  It  is  the  most 
serious  economic  mistake,  to  use  anything  special,  where  standard 
can  be  made  to  serve  satisfactorily,  and  this  mistake  is  the  most 
frequent  in  modern  electrical  engineering,  due  to  the  innate 
individualism  of  the  engineers. 

257.  However,  while  existing  standard  types  of  apparatus  are 
economically  preferable  wherever  they  can  be  used,  it  is  obvious 
that  with  the  rapid  expansion  of  the  industry,  new  types  of 
apparatus  will  be  developed,  introduced  and  become  standard, 
to  meet  new  conditions,  and  for  this  reason,  as  stated  above,  a 
knowledge  of  the  entire  known  field  of  apparatus  is  necessary 
to  the  engineer. 


CONCLUSION  475 

Most  of  the  less-known  and  less-used  types  of  apparatus  have 
been  discussed  in  the  preceding,  and  a  comprehensive  list  of 
them  is  given  in  Chapter  XXIII,  together  with  their  definitions 
and  short  characterization. 

While  electric  machines  are  generally  divided  into  induction 
machines,  synchronous  machines  and  commutating  machines, 
this  classification  becomes  difficult  in  considering  all  known 
apparatus,  as  many  of  them  fall  in  two  or  even  all  three  classes, 
or  are  intermediate,  or  their  inclusion  in  one  class  depends  on  the 
particular  definition  of  this  class. 

Induction  machines  consist  of  a  magnetic  circuit  inductively 
related,  that  is,  interlinked  with  two  sets  of  electric  circuits, 
which  are  movable  with  regards  to  each  other. 

They  thus  differ  from  transformers  or  in  general  stationary 
induction  apparatus,  in  that  the  electric  circuits  of  the  latter  are 
stationary  with  regards  to  each  other  and  to  the  magnetic  circuit. 

In  the  induction  machines,  the  mechanical  work  thus  is  pro- 
duced— or  consumed,  in  generators — by  a  disappearance  or 
appearance  of  electrical  energy  in  the  transformation  between 
the  two  sets  of  electric  circuits,  which  are  movable  with  regards 
to  each  other,  'and  of  which  one  may  be  called  the  primary  cir- 
cuit, the  other  the  secondary  circuit.  The  magnetic  field  of  the 
induction  machine  inherently  must  be  an  alternating  field 
(usually  a  polyphase  rotating  field)  excited -»  by  alternating 
currents. 

Synchronous  machines  are  machines  in  which  the  frequency  of 
rotation  has  a  fixed  and  rigid  relation  to  the  frequency  of  the 
supply  voltage. 

Usually  the  frequency  of  rotation  is  the  same  as  the  frequency 
of  the  supply  voltage:  in  the  standard  synchronous  machine, 
with  direct-current  field  excitation. 

The  two  frequencies,  however,  may  be  different:  in  the  double 
synchronous  generator,  the  frequency  of  rotation  is  twice  the 
frequency  of  alternation;  in  the  synchronous-induction  machine, 
it  is  a  definite  percentage  thereof;  so  also  it  is  in  the  induction 
machine  concatenated  to  a  synchronous  machine,  etc. 

Commutating  machines  are  machines  having  a  distributed 
armature  winding  connected  to  a  segmental  commutator. 

They  may  be  direct-current  or  alternating-current  machines. 

Unipolar  machines  are  machines  in  which  the  induction  is 
produced  by  the  constant  rotation  of  the  conductor  through  a 
constant  and  continuous  magnetic  field. 


476  ELECTRICAL  APPARATUS 

The  list  of  machine  types  and  their  definitions,  given  in 
Chapter  XXIII,  shows  numerous  instances  of  machines  belong- 
ing into  several  classes. 

The  most  common  of  these  double  types  is  the  converter,  or 
synchronous  commutating  machine. 

Numerous  also  are  the  machines  which  combine  induction- 
machine  and  synchronous-machine  characteristics,  as  the  double 
synchronous  generator,  the  synchronous-induction  motor  and 
generator,  etc. 

The  synchronous-induction  machine  comprising  a  polyphase 
stator  and  polyphase  rotor  connected  in  parallel  with  the  stator 
through  a  commutator,  is  an  induction  machine,  as  stator  and 
rotor  are  inductively  related  through  one  alternating  magnetic 
circuit;  it  is  a  synchronous  machine,  as  its  frequency  is  definitely 
fixed  by  the  speed  (and  ratio  of  turns  of  stator  and  rotor),  and 
it  also  is  a  commutating  machine. 

Thus  it  is  an  illustration  of  the  impossibility  of  a  rigid  classi- 
fication of  all  the  machine  types. 


INDEX 


Also  see  alphabetical  list  of  apparatus  in  Chapter  XXIII. 


Acyclic,  see  Unipolar. 

Adjustable  speed  polyphase  motor, 

321,  378 
Alexanderson  very  high  frequency 

inductor  alternator,  279 
Amplifier,  281 
Arc  rectifier,  248 
Armature    reaction    of    regulating 

pole  converter,  426,  437 
of  unipolar  machine,  457 


Balancer,  phase,  228 
Battery  charging  rectifier,  244 
Brush  arc  machine  as  quarterphase 
rectifier,  244,  254 


Capacity   storing   energy   in   phase 

conversion,  212 

Cascade  control,  see  Concatenation. 
Coil   distribution    giving    harmonic 
torque  in  induction  motor, 
151 

Commutating  e.m.f.  in  rectifier,  239 
field,    singlephase    commutator 

motor,  355,  359 

machine,  concatenation  with  in- 
duction motor,  55,  78 
pole  machine,  472 
poles,  singlephase  commutator 

motor,  358 
Commutation     current,      repulsion 

motor,  392 
series   repulsion   motor,    400, 

404 
factor,  repulsion  motor,  392 


Commutation  factor  of  series  repul- 
sion motor,  415 
of    regulating    pole    converter, 

426,  437 

of  series  repulsion  motor,  403 
of      singlephase      commutator 

motor,  347 
Commutator  excitation  of  induction 

motor,  54,  89 
induction  generator,  200 
leads,  singlephase  commutator 

motor,  351 

motors,  singlephase,  331 
Compensated  series  motor,  372 
Compensating  winding,  singlephase 
commutator    motor,     336, 
338 
Concatenation  of  induction  motors, 

14,  40 
Condenser   excitation   of  induction 

motor  secondary,  55,  84 
singlephase    induction    motor, 

120 
speed     control     of     induction 

motor,  13,  16 

Contact  making  rectifier,  245 
Cumulative  oscillation  of  synchro- 
nous machine,  299 

D 

Deep  bar  rotor  of  induction  motor, 
11- 

Delta  connected  rectifier,  251 

Direct  current  in  induction  motor 
secondary,  54,  57 

Disc  type  of  unipolar  machine,  454 

Double  squirrel  cage  induction 
motor,  29 

Double  synchronous  induction  gen- 
erator, 191,  199,  201 

Drum  type  of  unipolar  machine,  454 


477 


478 


INDEX 


E 


Eddy  current  starting  device  of  in- 
duction motor,  8 
in  unipolar  machine,  456 

P^ickemeyer  high  frequency  inductor 
alternator,  280 


Flashing  of  rectifier,  249 
Frequency  converter,  176 

pulsation,    effect    in    induction 

motor,  131 
Full  wave  rectifier,  245 


0 


General  alternating  current  motor, 
300 

Generator  regulation  affecting  induc- 
tion motor  stability,  137 


11 


Half  wave  rectifier,  245 

Harmonic  torque  of  induction  motor, 

144 

Heyland  motor,  92 
Higher  harmonic  torques  in  induc- 
tion motor,  144 
Homopolar,  see  Unipolar. 
Hunt  motor,  49 
Hunting,  see  Surging. 
Hysteresis  generator,  169 
motor,  168 

starting    device    of    induction 
motor,  5 


Independent  phase  rectifier,  251* 
Inductance  storing  energy  in  phase 

conversion,  212 

Inductive  compensation  of  single- 
phase  commutator  motor, 
343 

devices  starting  singlephase  in- 
duction motor,  97,  111 


Inductive  excitation  of  singlephase 

commutator  motor,  343 
Induction  frequency  converter,  191 
generator,  473 

motor  inductor  frequency  con- 
verter, 284 

phase  balancer  stationary,  228 
phase  converter,  220 
Inductor  machines,  274 
Interlocking  pole  type  of  machine, 

286 

Internally    concatenated    induction 
motor,  41,  49 


Lead  of  current  produced  by  lagging 
field  of  singlephase  com- 
mutator motor,  366 

Leblanc's  rectifier,  256 

Load  and  stability  of  induction 
motor,  132 

Low  frequency  exciter  of  induction 
generator,  199,  203 


Magneto  commutation,  285 
inductor  machine,  285 
Mechanical  starting  of  singlephase 

induction  motor,  96 
Mercury  arc  rectifier,  247 
Meter,  unipolar,  458 
Momentum  storing  energy  in  phase 

conversion,  212 
Monocyclic  devices,  214 

starting   singlephase   induction 

motor,  98,  117 
Motor  converter,  192 
Multiple  speed  induction  motor,  14, 

20 
Multiple    squirrel    cage    induction 

motor,  11,  27 


Open  circuit  rectifier,  237 
Over  compensation,  singlephase  com- 
mutator motor,  418 


INDEX 


479 


Permutator,  257 
Phase  balancer,  228 

control    by    polyphase    shunt 

motor,  324 

by     commutating     machine 
with  lagging  field  flux,  370 
conversion,  212 
converter   starting   singlephase 

induction  motor,  98 
splitting  devices  starting  single- 
phase  induction  motor,  97, 
103 
Polyphase    excitation    of    inductor 

alternator,  283 
induction  motor,  307 
rectifier,  250 
series  motor,  327 
shunt  motor,  319 
Position  angle  of  brushes  affecting 

converter  ratio,  422 
Power  factor  compensation  by  com- 
mutator motor,  379 
of  frequency  converter,  178,  184 
Pyroelectric  speed  control  of  induc- 
tion motor,  14 


Quarterphase  rectifier,  251 


R 


Reaction  converter,  264 
machine,  260 

Rectifier,  synchronous,  234 

Regulating  pole  converter,  422 

Regulation  coefficient  of  system  and 
induction  motor  stability, 
140 
of  induction  motor,  123 

Regulator,   voltage-,  magneto  com- 
mutation, 285 

Repulsion  motor,  343,  373,  385 

starting   of  singlephase   induc- 
tion motor,  97 

Resistance  speed  control  of  induc- 
tion motor,  12 


Reversing  rectifier,  245 
Ring  connected  rectifier,  251 
Rotary  terminal  singlephase  induc- 
tion motor,  172 


S 


Secondary   excitation   of   induction 

motor,  52 

Self  induction  of  commutation,  420 
Semi-inductor  type  of  machine,  286 
Series  repulsion  motor,  343,  374,  397 
Shading  coil  starting  device,  112 
Short  circuit  rectifier,  237 
Shunt  resistance  of  rectifier,  235 
and    series    motor    starting    of 
singlephase      induction 
motor,  96 

Singlephase  commutator  motor,  331 
generation,  212,  229 
induction  motor,  93,  314 

self  starting  by   rotary  ter- 
minals, 172 
Six-phase  rectifier,  253 

regulating  pole  converter,  446 
Split  pole  converter,  see  Regulating 

pole  converter. 
Square,  monocycle,  216 
Stability  coefficient  of  induction 

motor,  138 

of  system    containing  induc- 
tion motor,  141 
Stability   of   induction   motor   and 

generator  regulation,  137 
limit  of  rectifier,  249 
and  load  of  induction  motor,  132 
Stanley  inductor  alternator,  275 
Star  connected  rectifier,  251 
Surging  of  synchronous  machine,  288 
Synchronizing  induction   motor  on 

common  rheostat,  159 
Synchronous    exciter    of    induction 

motor,  72 

frequency  converter,  191 
induction  generator,  191,  194 
induction    generator  with  low 
frequency  exciter,  199,  203 
induction  motor,  166 
as  reaction  machine,  264 


480 


INDEX 


Synchronous  machines,  surging,  288 
motor,   concatenation  with  in- 
duction motor,  54,  71 
phase  balancer,  228 
phase  converter,  227 
rectifier,  234 


U 

Unipolar  induction,  452 
machines,  450 
motor  meter,  458 


Tandem  control,  see  Concatenation. 
Temperature     starting     device     of 

induction  motor,  2 
Third    harmonic    wave    controlling 

converter  ratio,  432 
Thomson-Houston    arc  machine  as 

three-phase    rectifier,   244, 

255 
Three-phase  rectifier,  251 

regulating  pole  converter,  445 
transformer,  472 

Transformer,  general  alternating,  176 
Triangle,  monocyclic,  216 
Triple  squirrel  cage  induction  motor, 

34 


Variable  ratio  converter,  see  Regu- 
lating pole  converter. 


W 


Wave     shape     affecting     converter 

ratio,  430 
harmonics   giving  induction 

motor  torque,  145 
Winter-Eichberg  motor,  380 


Y  connected  rectifier,  251 


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